#7186724

order	512
nilpotency class	3
isoclinism	isoclinic class of #7021351
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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1664 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=h,c^2=g,d^2=fi,e^2=hi,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

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, 7093920);