#4404672

order	512
nilpotency class	2
isoclinism	isoclinic class of #4379698
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

1366 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)(9,10,13,16)(11,14,12,15)(17,18)(19,25)(20,28)(21,22)(23,26)(24,30)(27,31)(29,32)(33,34)(35,42)(36,40)(37,39)(38,41)(43,48)(44,47)(45,46);(18,22)(19,26)(24,28)(27,29)(33,35,39,36)(34,40,37,42)(38,43,46,44)(41,47,45,48);(9,10)(11,14)(12,15)(13,16)(17,19)(18,23)(20,29)(21,26)(22,25)(24,32)(27,30)(28,31)(33,36,39,35)(34,40,37,42)(38,43,46,44)(41,48,45,47);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,16,15)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,30)(29,32)(33,37,39,34)(35,40,36,42)(38,45,46,41)(43,47,44,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,15,16,14)(17,20)(18,24)(19,27)(21,30)(22,28)(23,31)(25,32)(26,29)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=fh,c^2=fh,d^2=fg,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=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, 4311868);