#382945

order	512
nilpotency class	3
isoclinism	isoclinic class of #188768
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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×#383
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

934 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)(11,16)(12,25,28,36)(13,19)(14,20)(15,18)(17,27,35,29)(21,34,37,40)(22,32)(23,33)(24,31)(26,39,38,30);(2,5)(6,8)(9,11,13,22,15,24,14,23)(10,16,19,32,18,31,20,33)(12,26,27,37,28,38,29,21)(17,34,25,39,35,40,36,30);(1,3)(2,6)(4,7)(5,8)(9,12)(10,17)(11,21,24,37)(13,29)(14,27)(15,28)(16,30,31,39)(18,35)(19,36)(20,25)(22,38,23,26)(32,40,33,34);(3,7)(6,8)(10,18)(16,31)(17,35)(19,20)(25,36)(30,39)(32,33)(34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290141);