#7174129

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1319 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:

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=gi,d^2=i,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=dhi,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, 7081325);