#7204226

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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	#35772	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1540 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)(3,5)(4,6)(7,8)(9,10,13,15)(11,14,12,16)(17,18)(19,27)(20,23)(21,25)(22,24)(26,30)(28,32)(29,31);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,14)(13,15)(17,19)(18,23)(20,21)(22,28)(24,31)(25,27)(26,32)(29,30);(3,7)(5,8)(9,11,13,12)(10,14,15,16)(18,24)(22,30)(23,31)(25,26)(27,32)(28,29);(2,6)(3,7)(9,12,13,11)(10,14,15,16)(17,20,21,19)(18,23,25,27)(22,29,30,28)(24,31,26,32);(3,7)(5,8)(10,15)(14,16)(18,25)(19,20)(24,26)(28,29)

Polycyclic presentation:

2^9|a^2=fi,c^2=fi,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,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, 7111422);