#5164692

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

1322 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)(11,15)(12,13)(14,16)(18,22)(19,25)(20,28)(30,32)(33,34,37,38)(35,41,42,39)(36,44,45,40)(43,46,48,47);(9,11,13,16)(10,14,12,15)(18,22)(20,28)(23,26)(27,31)(34,38)(35,42)(36,45)(46,47);(9,10,13,12)(11,14,16,15)(17,18,21,22)(19,26,25,23)(20,24,28,29)(27,32,31,30)(33,34)(35,39)(36,44)(37,38)(40,45)(41,42)(43,47)(46,48);(1,2)(3,5)(4,6)(7,8)(9,12,13,10)(11,14,16,15)(17,19,21,25)(18,23,22,26)(20,27,28,31)(24,30,29,32)(33,35,37,42)(34,39,38,41)(36,43,45,48)(40,46,44,47);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,14)(13,16)(17,20)(18,24)(19,27)(21,28)(22,29)(23,30)(25,31)(26,32)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48)

Polycyclic presentation:

2^9|a^2=fhi,b^2=gh,c^2=fh,d^2=g,e^2=gi,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, 5071888);