#937617

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2726 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,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, 844813);