#4526131

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

1423 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=hi,d^2=i,e^2=fgh,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, 4433327);