#5598838

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,c^2=f,d^2=fgh,e^2=hi,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=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, 5506034);