#7145268

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

1425 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)(11,15)(14,16)(17,18)(19,25)(20,27)(21,23)(22,26)(24,28)(29,32)(30,31);(1,3,5,8)(2,6,4,7)(10,13)(11,15)(18,23)(24,31)(25,32)(26,27);(1,4,5,2)(3,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,26)(27,32);(3,8)(6,7)(10,13)(14,16)(17,19)(18,25)(20,30)(21,29)(22,28)(23,32)(24,26)(27,31);(9,11,12,15)(10,14,13,16)(17,20,21,22)(18,26,23,27)(19,28,29,30)(24,25,31,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=dfi,e^b=eh,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, 7052464);