#5783434

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1459 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fghi,c^2=i,d^2=g,e^2=g,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, 5690630);