#767102

order	512
nilpotency class	2
isoclinism	isoclinic class of #763332
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2130 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 674298);