#2800293

order	512
nilpotency class	2
isoclinism	isoclinic class of #2760365
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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1362 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=fg,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,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, 2707489);