#4226792

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
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

1316 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fg,c^2=ghi,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,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, 4133988);