#2746172

order	512
nilpotency class	2
isoclinism	isoclinic class of #2735949
automorphism	Aut(G) order 2097152
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

1572 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=h,d^2=fi,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, 2653368);