#4928699

order	512
nilpotency class	2
isoclinism	isoclinic class of #4861280
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

1292 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=fhi,c^2=ghi,d^2=fgh,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,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, 4835895);