#5164321

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

1300 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)(9,10,15,16)(11,12,13,14)(18,22)(19,25)(20,28)(30,32)(33,34,38,35)(36,39,43,41)(37,42,45,40)(44,47,48,46);(9,11,15,13)(10,12,16,14)(18,22)(20,28)(23,26)(27,31)(34,35)(36,43)(40,42)(44,48);(9,12,15,14)(10,11,16,13)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31)(33,35,38,34)(36,39,43,41)(37,40,45,42)(44,47,48,46);(1,2)(3,5)(4,6)(7,8)(9,13,15,11)(10,12,16,14)(17,19,21,25)(18,23,22,26)(20,27,28,31)(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)(9,14,15,12)(10,11,16,13)(17,20,21,28)(18,24,22,29)(19,27,25,31)(23,30,26,32)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=fgh,c^2=gh,d^2=g,e^2=fi,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, 5071517);