#2667036

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
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

1557 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)(3,7)(10,14)(12,15)(17,18,21,19)(20,23,24,22)(25,26)(27,33)(28,36)(29,34)(30,31)(32,37)(35,39)(38,40);(3,7)(6,8)(10,14)(11,16)(17,18,21,19)(20,22,24,23)(26,31)(27,34)(28,37)(38,39);(1,2,5,4)(3,6,7,8)(9,10)(11,15)(12,16)(13,14)(17,19,21,18)(20,23,24,22)(25,27)(26,29)(28,35)(30,34)(31,33)(32,39)(36,38)(37,40);(1,3)(2,6)(4,8)(5,7)(9,11,13,16)(10,15,14,12)(17,20,21,24)(18,22,19,23)(25,28)(26,32)(27,35)(29,39)(30,37)(31,36)(33,38)(34,40);(1,4,5,2)(3,8,7,6)(9,12,13,15)(10,16,14,11)(25,29)(26,27)(28,38)(30,33)(31,34)(32,40)(35,36)(37,39)

Polycyclic presentation:

2^9|a^2=fghi,b^2=fghi,c^2=fi,d^2=gi,e^2=fg,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, 2574232);