#508037

order	512
nilpotency class	4
isoclinism	isoclinic class of #508036
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#29594
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#60471	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #3573 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

963 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 963 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

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, 415233);