#6074362

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

1292 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:

(2,5)(3,7)(9,10,14,12)(11,15,13,16)(18,22)(19,25)(24,29)(27,31)(33,34)(35,39)(36,38)(37,40)(41,43)(42,46)(44,45)(47,48);(2,5)(6,8)(9,11,14,13)(10,15,12,16)(18,22)(23,26)(24,29)(30,32)(34,36)(35,41)(37,45)(46,48);(9,12,14,10)(11,15,13,16)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,35,38,41)(34,39,36,43)(37,42,45,47)(40,46,44,48);(1,2,4,5)(3,6,7,8)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,36,38,34)(35,39,41,43)(37,44,45,40)(42,46,47,48);(1,3)(2,6)(4,7)(5,8)(9,13,14,11)(10,15,12,16)(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=fgh,b^2=fgh,c^2=h,d^2=i,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, 5981558);