#4980392

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
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

1592 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=i,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

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, 4887588);