#937758

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2720 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=ghi,d^2=g,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, 844954);