#4445864

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
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

1360 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)(6,8)(9,10,14,13)(11,16,15,12)(17,18)(19,25)(20,28)(21,22)(23,27)(24,30)(26,31)(29,32)(33,34,38,36)(35,41,42,39)(37,44,45,40)(43,46,48,47);(9,10,14,13)(11,12,15,16)(18,22)(23,25)(24,28)(31,32)(35,42)(39,41)(43,48)(46,47);(9,11)(10,12)(13,16)(14,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,35,38,42)(34,39,36,41)(37,43,45,48)(40,46,44,47);(1,2)(3,6)(4,5)(7,8)(9,12,14,16)(10,15,13,11)(17,19,21,27)(18,23,22,25)(20,29,30,26)(24,32,28,31)(33,36,38,34)(35,41,42,39)(37,44,45,40)(43,47,48,46);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,16,15,12)(17,20,21,30)(18,24,22,28)(19,26,27,29)(23,31,25,32)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=g,b^2=fgh,c^2=fh,d^2=fg,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4353060);