#2602588

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

2192 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:

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

Polycyclic presentation:

2^9|b^2=fg,c^2=fh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,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, 2509784);