#212638

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₈	C₂×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=e,d^2=g,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ci

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 119834);