#208494

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

807 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)(9,10,14,12)(11,17)(13,16,18,22)(15,23)(19,20)(21,24)(25,26,29,34,30,33,28,32)(27,35,36,31,38,39,37,40);(9,11,18,20,14,21,13,23)(10,15,22,17,12,19,16,24)(25,27)(26,31)(28,36)(29,37)(30,38)(32,39)(33,40)(34,35);(2,5)(6,8)(9,12,14,10)(11,19,21,15)(13,22,18,16)(17,20,24,23)(25,28,30,29)(26,32,33,34)(27,36,38,37)(31,39,40,35);(1,3,4,7)(2,6,5,8)(10,12)(15,19)(16,22)(17,24)(26,33)(31,40)(32,34)(35,39)

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=h,d^2=f,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^b=eh,g^b=gh,d^c=dhi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 115690);