#477324

order	512
nilpotency class	3
isoclinism	isoclinic class of #476707
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

850 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 850 conjugacy classes of subgroups — too many to draw. Only its 170 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:

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

Polycyclic presentation:

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

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, 384520);