#204459

order	512
nilpotency class	3
isoclinism	isoclinic class of #24278
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#390
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1116 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:

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

Polycyclic presentation:

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

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, 111655);