#478200

order	512
nilpotency class	3
isoclinism	isoclinic class of #476144
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

914 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2,4,3)(5,6)(7,17)(8,22)(9,27)(10,32)(11,16)(12,20)(13,25)(14,31)(15,34)(18,35)(19,49)(21,46)(23,56)(24,41)(26,43)(28,45)(29,38)(30,42)(33,39)(36,62)(37,54)(40,52)(44,48)(47,64)(50,60)(51,68)(53,55)(57,66)(58,59)(61,65)(63,67);(1,2)(3,4)(5,7,10,20,11,21,34,54)(6,12,15,17,16,37,32,46)(8,23,25,53,26,18,33,48)(9,28,30,62,31,19,50,52)(13,35,22,55,39,56,43,44)(14,40,42,49,27,36,60,45)(24,47,29,51,58,57,63,61)(38,66,41,68,67,64,59,65);(1,3,4,2)(5,8)(6,13)(7,18)(9,29)(10,33)(11,26)(12,35)(14,41)(15,43)(16,39)(17,44)(19,51)(20,53)(21,23)(22,32)(24,30)(25,34)(27,59)(28,61)(31,63)(36,65)(37,56)(38,42)(40,68)(45,64)(46,55)(47,52)(48,54)(49,66)(50,58)(57,62)(60,67);(2,3)(5,9,10,30,11,31,34,50)(6,14,15,42,16,27,32,60)(7,19,20,52,21,28,54,62)(8,24,25,29,26,58,33,63)(12,36,17,45,37,40,46,49)(13,38,22,41,39,67,43,59)(18,47,48,51,23,57,53,61)(35,64,55,65,56,66,44,68)

Polycyclic presentation:

2^9|a^2=efi,b^2=f,c^2=efi,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385396);