#500681

order	512
nilpotency class	4
isoclinism	isoclinic class of #500681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33519	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1149 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 407877);