#4762835

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

1380 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)(11,12)(13,14)(15,16)(18,22)(19,25)(20,27)(30,32)(33,34,39,36)(35,40,37,43)(38,44,46,41)(42,48,45,47);(10,14)(11,16)(18,22)(19,25)(24,29)(26,31)(33,35,39,37)(34,40,36,43)(38,42,46,45)(41,47,44,48);(10,14)(12,15)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,36,39,34)(35,40,37,43)(38,44,46,41)(42,47,45,48);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,15,14,12)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32)(33,37,39,35)(34,40,36,43)(38,45,46,42)(41,47,44,48);(1,3,4,7)(2,6,5,8)(9,12,13,15)(10,16,14,11)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)(33,38)(34,41)(35,42)(36,44)(37,45)(39,46)(40,47)(43,48)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgh,c^2=h,d^2=gh,e^2=fghi,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, 4670031);