#7198834

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1372 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)(3,6)(4,5)(7,8)(10,11)(12,16)(17,18,22,28)(19,26,25,24)(20,31,32,30)(21,29,23,27);(2,5)(3,7)(9,10)(11,13)(12,15)(14,16)(17,19,22,25)(18,24,28,26)(20,27,32,29)(21,31,23,30);(9,11,13,10)(12,15,16,14)(17,20,21,24,22,32,23,26)(18,25,27,30,28,19,29,31);(1,3,4,7)(2,6,5,8)(10,11)(14,15)(17,19)(18,26)(20,27)(21,31)(22,25)(23,30)(24,28)(29,32);(9,12,13,16)(10,14,11,15)(17,21,22,23)(18,27,28,29)(19,30,25,31)(20,26,32,24)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7106030);