#5117882

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

1354 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=fgi,c^2=gh,d^2=fi,e^2=h,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, 5025078);