#2636736

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
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

1442 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=fg,c^2=hi,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,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, 2543932);