#5980999

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

1469 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=h,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, 5888195);