#2722169

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1713 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=f,d^2=f,e^2=ghi,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, 2629365);