#5806122

order	512
nilpotency class	2
isoclinism	isoclinic class of #5605735
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

1302 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)(3,6)(4,5)(7,8)(9,10,13,11)(12,15,16,14)(17,18,22,19)(20,25,28,23)(21,24,30,26)(27,32,29,31)(33,34,38,36)(35,39,41,42)(37,44,45,40)(43,48,47,46);(3,7)(6,8)(9,10,13,11)(12,14,16,15)(18,19)(23,25)(24,26)(31,32)(35,41)(39,42)(43,47)(46,48);(2,5)(6,8)(9,11,13,10)(12,15,16,14)(17,19,22,18)(20,23,28,25)(21,26,30,24)(27,31,29,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,2)(3,6)(4,5)(7,8)(9,12)(10,14)(11,15)(13,16)(17,20,22,28)(18,23,19,25)(21,29,30,27)(24,32,26,31)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,3)(2,6)(4,7)(5,8)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=i,d^2=h,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=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, 5713318);