#7010324

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#28354	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1441 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)(17,18,20,25,21,22,30,24)(19,26,29,32,28,23,27,31);(1,3,5,8)(2,6,4,7)(9,10)(11,15)(12,14)(13,16)(18,22)(23,26)(24,25)(31,32);(1,2,5,4)(3,6,8,7)(9,11)(10,13)(12,16)(14,15)(17,18)(19,26)(20,24)(21,22)(23,28)(25,30)(27,32)(29,31);(1,4,5,2)(3,6,8,7)(11,16)(13,15)(17,19,21,28)(18,23,22,26)(20,29,30,27)(24,32,25,31);(10,14)(13,15)(17,20,21,30)(18,24,22,25)(19,27,28,29)(23,31,26,32)

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,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, 6917520);