#148088

order	512
nilpotency class	4
isoclinism	isoclinic class of #13304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×D₈
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂³	C₄×D₈
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

718 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=hi,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=eh,f^b=fh,h^b=hi,e^c=eh,f^c=fh,h^c=hi

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, 55284);