#1226620

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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

1750 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)(3,7)(4,8)(5,6)(10,13)(11,15)(17,18,21,24)(19,20,23,22)(26,28)(30,31);(1,2,5,6)(3,4,8,7)(17,19,21,23)(18,22,24,20);(2,6)(3,8)(9,10)(11,14)(12,13)(15,16)(17,20,21,22)(18,23,24,19)(25,26,29,28)(27,30,32,31);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,20)(18,19)(21,22)(23,24)(25,27,29,32)(26,30,28,31);(1,4)(2,3)(5,7)(6,8)(9,11,12,15)(10,14,13,16)(17,19)(18,20)(21,23)(22,24)(25,28,29,26)(27,31,32,30)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=gi,d^2=fghi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1133816);