#7160721

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1417 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=fgi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 7067917);