#7022072

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	#25775	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1716 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:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=g,d^2=f,e^2=i,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, 6929268);