#941174

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

1944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(2,5)(6,8)(9,10,13,11)(12,15,16,14)(19,23)(21,24)(25,26,30,31)(27,33,35,28)(29,37,38,32)(34,36,40,39);(3,7)(6,8)(17,18,20,22)(19,21,23,24)(26,31)(28,33)(32,37)(36,39);(1,2,4,5)(3,6,7,8)(9,11,13,10)(12,15,16,14)(17,18)(19,21)(20,22)(23,24)(25,27,30,35)(26,28,31,33)(29,34,38,40)(32,36,37,39);(1,3)(2,6)(4,7)(5,8)(9,12)(10,14)(11,15)(13,16)(17,19,20,23)(18,21,22,24)(25,28)(26,27)(29,36)(30,33)(31,35)(32,34)(37,40)(38,39);(25,29,30,38)(26,32,31,37)(27,34,35,40)(28,36,33,39)

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,c^2=ghi,d^2=h,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, 848370);