#4751011

order	512
nilpotency class	2
isoclinism	isoclinic class of #4597856
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)(11,15)(14,16)(17,18,22,20)(19,25,26,23)(21,24,30,28)(27,32,29,31)(35,41)(36,44)(39,43)(40,45);(10,13)(14,16)(18,20)(19,26)(24,28)(27,29)(33,34,38,37)(35,39,41,43)(36,45,44,40)(42,48,47,46);(9,10,12,13)(11,14,15,16)(17,19,22,26)(18,23,20,25)(21,29,30,27)(24,32,28,31)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(1,2)(3,6)(4,5)(7,8)(17,20,22,18)(19,25,26,23)(21,28,30,24)(27,32,29,31)(33,36,38,44)(34,40,37,45)(35,42,41,47)(39,46,43,48);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,21,22,30)(18,24,20,28)(19,27,26,29)(23,31,25,32)(33,37,38,34)(35,43,41,39)(36,45,44,40)(42,48,47,46)

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=h,d^2=g,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef

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, 4658207);