#7020692

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#44871	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1773 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=fgi,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=eh,d^c=dhi,e^c=ei,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, 6927888);