#6130904

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1460 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,4)(3,7,8,6)(9,10,12,14)(11,15,16,13)(18,22)(20,28)(23,26)(27,31)(33,34,38,36)(35,40,43,39)(37,44,46,41)(42,47,45,48);(2,4)(3,8)(11,16)(13,15)(18,22)(19,25)(24,29)(27,31)(33,35,38,43)(34,39,36,40)(37,45,46,42)(41,48,44,47);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,35)(34,40)(36,39)(37,42)(38,43)(41,48)(44,47)(45,46);(1,4,5,2)(3,7,8,6)(9,10)(11,15)(12,14)(13,16)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36,38,34)(35,40,43,39)(37,44,46,41)(42,48,45,47);(1,3)(2,6)(4,7)(5,8)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,37,38,46)(34,41,36,44)(35,42,43,45)(39,47,40,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=gh,c^2=fhi,d^2=hi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6038100);