#368934

order	512
nilpotency class	3
isoclinism	isoclinic class of #196299
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1212 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 276130);