#7045353

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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	#25106	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1436 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=fgi,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,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, 6952549);