#7027387

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	#23850	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1656 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,e^2=f,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, 6934583);