#182894

order	512
nilpotency class	3
isoclinism	isoclinic class of #17577
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1120 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=h,f^2=h,g^2=i,b^a=be,c^a=ch,e^a=eh,g^a=ghi,d^b=dh,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90090);