#7207326

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	#25762	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1697 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=efi,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7114522);