#416601

order	512
nilpotency class	3
isoclinism	isoclinic class of #394180
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=fi,d^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323797);