#767103

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1886 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)(3,7)(4,5)(6,8)(10,13)(11,15)(18,20)(19,23)(25,26,30,31)(27,29,28,32);(2,5)(6,7)(9,10,12,13)(11,14,15,16)(17,18,21,20)(19,24,23,22)(26,31)(27,28);(3,8)(6,7)(11,15)(14,16)(18,20)(22,24)(25,27,30,28)(26,32,31,29);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,23)(18,22,20,24)(25,28,30,27)(26,32,31,29);(1,3,4,8)(2,6,5,7)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,24,23,22)(25,29,30,32)(26,27,31,28)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=h,e^2=f,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, 674299);