#5073733

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

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)(3,7)(4,5)(6,8)(9,10,13,11)(12,15,16,14)(18,22)(20,28)(23,25)(26,32)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,46,48,47);(2,5)(3,8)(9,11,13,10)(12,15,16,14)(18,22)(20,28)(23,25)(26,32)(34,38)(35,42)(36,45)(46,47);(2,5)(6,7)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34)(35,39)(36,44)(37,38)(40,45)(41,42)(43,47)(46,48);(1,3,4,8)(2,6,5,7)(9,12)(10,14)(11,15)(13,16)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,2)(3,6)(4,5)(7,8)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(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=fi,c^2=fg,d^2=g,e^2=fgh,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, 4980929);