#3655226

order	512
nilpotency class	2
isoclinism	isoclinic class of #3332717
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

1378 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=g,c^2=fg,d^2=fgh,e^2=ghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 3562422);