#7147484

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993979
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	C₂×#496	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1541 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,d^2=f,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7054680);