#4341213

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1698 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=gi,d^2=fgh,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,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, 4248409);