#843209

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1882 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgi,c^2=g,d^2=fh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 750405);