#949547

order	512
nilpotency class	2
isoclinism	isoclinic class of #917169
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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=fgh,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, 856743);