#4597306

order	512
nilpotency class	2
isoclinism	isoclinic class of #4465744
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

1296 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,c^2=ghi,d^2=fgh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df

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, 4504502);