#190262

order	512
nilpotency class	3
isoclinism	isoclinic class of #190208
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1053 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)(3,5)(4,6)(7,8)(9,10,14,20,15,19,13,18)(11,21,24,31,25,16,23,32)(12,26,27,33,28,34,29,17)(22,36,37,35,39,40,38,30);(2,5)(6,7)(9,11,15,25)(10,16,19,21)(12,22,28,39)(13,24,14,23)(17,30,33,35)(18,32,20,31)(26,40,34,36)(27,38,29,37);(2,6)(4,8)(5,7)(9,12)(10,17,19,33)(11,22)(13,27)(14,29)(15,28)(16,30,21,35)(18,26,20,34)(23,37)(24,38)(25,39)(31,36,32,40);(9,13,15,14)(10,18,19,20)(11,23,25,24)(12,27,28,29)(16,31,21,32)(17,26,33,34)(22,37,39,38)(30,36,35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ei,d^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,g^b=gh,f^c=fi,g^c=gh

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, 97458);