#6640379

order	512
nilpotency class	3
isoclinism	isoclinic class of #6636868
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	#35171	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1510 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,5)(4,6)(7,8)(10,13)(11,15)(14,16)(17,18,22,25)(19,26,28,23)(20,24,21,30)(27,32,29,31);(1,3)(2,5)(4,7)(6,8)(9,10,12,13)(11,15,16,14)(17,19)(18,23)(20,27)(21,29)(22,28)(24,31)(25,26)(30,32);(3,7)(5,8)(9,11,13,15,12,16,10,14)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,31,26,32);(3,7)(5,8)(9,10,12,13)(11,15,16,14)(18,25)(19,28)(24,30)(27,29);(2,6)(5,8)(11,16)(14,15)(17,21,22,20)(18,24,25,30)(19,29,28,27)(23,31,26,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=fgi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=efh,g^a=gi,c^b=ch,d^b=dfh,d^c=di,e^c=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, 6547575);