#367291

order	512
nilpotency class	3
isoclinism	isoclinic class of #24384
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄²

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×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1164 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)(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)(9,11,13,21,14,22,15,23)(10,16,29,38,19,32,18,31)(12,25,27,37,26,20,28,36)(17,33,40,35,34,30,24,39);(6,8)(9,12,13,27,14,26,15,28)(10,17,18,24,19,34,29,40)(11,20,21,36,22,25,23,37)(16,30,31,35,32,33,38,39);(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=egh,c^2=f,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, 274487);