#5794008

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

1371 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:

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

Polycyclic presentation:

2^9|a^2=fghi,b^2=i,c^2=ghi,d^2=h,e^2=f,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, 5701204);