#946410

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2322 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 853606);