#6034977

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, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fgh,c^2=fi,d^2=f,e^2=fh,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, 5942173);