#392761

order	512
nilpotency class	3
isoclinism	isoclinic class of #32888
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1305 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,f^c=fi,g^d=gi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299957);