#5168377

order	512
nilpotency class	2
isoclinism	isoclinic class of #5035983
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

1298 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)(11,12)(14,15)(18,22)(19,25)(24,29)(27,31)(33,34)(35,41)(36,44)(37,38)(39,42)(40,45)(43,46)(47,48);(9,10,13,16)(11,14,12,15)(20,28)(24,29)(27,31)(30,32)(34,38)(35,42)(40,44)(43,48);(9,10)(11,14)(12,15)(13,16)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,34)(35,41)(36,44)(37,38)(39,42)(40,45)(43,46)(47,48);(1,2)(3,5)(4,6)(7,8)(9,11,13,12)(10,14,16,15)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,35)(34,39)(36,43)(37,42)(38,41)(40,46)(44,47)(45,48);(1,3,4,7)(2,5,6,8)(9,12,13,11)(10,15,16,14)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,36,37,45)(34,40,38,44)(35,43,42,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=fh,c^2=g,d^2=fgh,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,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, 5075573);