#369103

order	512
nilpotency class	3
isoclinism	isoclinic class of #24805
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1824 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=fg,d^2=ei,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^c=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276299);