#6786362

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#38035	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1530 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(17,18,22,26)(19,27,30,41)(20,24,32,38)(21,34,28,25)(23,35,37,31)(29,40,43,48)(33,45,42,39)(36,46,47,44);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(20,32)(24,38)(29,43)(33,42)(36,47)(39,45)(40,48)(44,46);(2,6)(3,8)(10,13)(14,16)(17,19,28,35,22,30,21,31)(18,23,34,41,26,37,25,27)(20,29,42,46,32,43,33,44)(24,36,45,48,38,47,39,40);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,22,32)(18,24,26,38)(19,29,30,43)(21,33,28,42)(23,36,37,47)(25,39,34,45)(27,40,41,48)(31,44,35,46);(10,13)(14,16)(19,30)(20,32)(23,37)(24,38)(27,41)(31,35)(33,42)(39,45)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6693558);