#4481897

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
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

1464 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)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,25)(20,28)(21,22)(23,26)(24,29)(27,30)(31,32)(33,34,38,39)(35,41,43,36)(37,40,45,46)(42,47,48,44);(3,7)(6,8)(9,11,14,12)(10,15,13,16)(18,22)(19,26)(20,29)(30,31)(34,39)(36,41)(37,45)(42,48);(1,2,4,5)(3,6,7,8)(9,12,14,11)(10,15,13,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,31)(29,32)(33,35,38,43)(34,36,39,41)(37,42,45,48)(40,44,46,47);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,15,12,16)(17,18)(19,25)(20,24)(21,22)(23,26)(27,31)(28,29)(30,32)(33,36)(34,35)(37,44)(38,41)(39,43)(40,42)(45,47)(46,48);(17,20,21,29)(18,24,22,28)(19,27,26,32)(23,30,25,31)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=gh,d^2=fg,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4389093);