#4526091

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

1368 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,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, 4433287);