#2747156

order	512
nilpotency class	2
isoclinism	isoclinic class of #2735949
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

1569 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fi,c^2=fgh,d^2=gi,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,d^c=df,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2654352);