#6854975

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1431 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)(9,10,13,12)(11,14,16,15)(17,18,22,26)(19,27,29,40)(20,24,31,38)(21,33,35,25)(23,34,37,30)(28,41,42,48)(32,45,44,39)(36,46,47,43);(1,3)(2,5)(4,8)(6,7)(9,11,13,16)(10,14,12,15)(20,31)(24,38)(28,42)(32,44)(36,47)(39,45)(41,48)(43,46);(9,10,13,12)(11,15,16,14)(17,19)(18,23)(20,28)(21,34)(22,29)(24,36)(25,40)(26,37)(27,33)(30,35)(31,42)(32,46)(38,47)(39,48)(41,45)(43,44);(2,6)(5,7)(17,20,22,31)(18,24,26,38)(19,28,29,42)(21,32,35,44)(23,36,37,47)(25,39,33,45)(27,41,40,48)(30,43,34,46);(3,8)(5,7)(9,12,13,10)(11,14,16,15)(19,29)(20,31)(23,37)(24,38)(27,40)(30,34)(32,44)(39,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=i,e^2=h,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,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6762171);