#7336549

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324468
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	#24434	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1635 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dhi,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7243745);