#6074711

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=fgh,d^2=fi,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, 5981907);