#7205670

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200002
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1500 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)(10,14)(11,12)(17,18,22,25)(19,23,21,29)(20,27,28,31)(24,30,32,26);(1,3)(2,5)(4,7)(6,8)(9,10)(11,15)(12,16)(13,14)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27);(3,7)(5,8)(10,14)(15,16)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29);(2,6)(3,7)(9,11,13,12)(10,15,14,16);(3,7)(5,8)(9,12,13,11)(10,15,14,16)(20,28)(24,32)(26,30)(27,31)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=f,e^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=eh,e^c=ei,g^c=gi

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, 7112866);