#6074409

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

Polycyclic presentation:

2^9|a^2=gh,b^2=fh,c^2=fgh,d^2=fi,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, 5981605);