#382542

order	512
nilpotency class	3
isoclinism	isoclinic class of #188766
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1882 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)(9,10,13,18,15,20,14,19)(11,16,22,31,23,32,29,36)(12,24,26,34,27,35,28,17)(21,37,38,33,25,40,39,30);(2,5)(6,8)(9,11)(10,16)(12,25)(13,22)(14,29)(15,23)(17,33)(18,31)(19,36)(20,32)(21,27)(24,40)(26,39)(28,38)(30,34)(35,37);(1,3)(2,6)(4,7)(5,8)(9,12,14,26,15,27,13,28)(10,17,19,24,20,34,18,35)(11,21,22,39,23,25,29,38)(16,30,31,40,32,33,36,37);(3,7)(6,8)

Polycyclic presentation:

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

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, 289738);