#7042084

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021363
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1712 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fhi,c^2=gi,d^2=fi,f^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,f^b=fi,d^c=dh,e^c=ei,e^d=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, 6949280);