#369082

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#3150	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1888 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fg,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^c=di

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, 276278);