#6122133

order	512
nilpotency class	2
isoclinism	isoclinic class of #5956507
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1402 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=ghi,e^2=fg,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, 6029329);