#829489

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2438 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fghi,d^2=fhi,e^2=g,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, 736685);