#5240749

order	512
nilpotency class	2
isoclinism	isoclinic class of #5211343
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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=fg,d^2=fg,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,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, 5147945);