#612406

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1834 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=f,d^2=fg,e^2=fhi,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^c=df

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 519602);