#643953

order	512
nilpotency class	2
isoclinism	isoclinic class of #635984
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1638 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)(3,4)(5,7)(6,8)(10,11)(14,15)(18,20)(19,21)(25,26,29,28)(27,31,32,30);(1,3,5,8)(2,6,7,4)(17,18,22,20)(19,24,21,23)(27,32)(30,31);(2,7)(3,8)(9,10,13,11)(12,14,16,15)(17,19,22,21)(18,23,20,24)(25,27)(26,30)(28,31)(29,32);(1,2)(3,6)(4,8)(5,7)(9,11,13,10)(12,15,16,14)(17,20,22,18)(19,23,21,24);(1,4,5,6)(2,8,7,3)(9,12,13,16)(10,14,11,15)(17,21,22,19)(18,23,20,24)(25,28,29,26)(27,31,32,30)

Polycyclic presentation:

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

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, 551149);