#6378561

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#36692	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1479 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cf,e^b=ei,d^c=di,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6285757);