#371351

order	512
nilpotency class	3
isoclinism	isoclinic class of #24464
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fh,d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^c=fi,h^c=hi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278547);