#502090

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#25803	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

754 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409286);