#369202

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1830 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)(5,6,8,7)(9,10)(11,23)(12,28)(13,32)(14,21)(15,20)(16,22)(17,26)(18,31)(19,33)(24,38)(25,40)(27,39)(29,37)(30,35)(34,36);(1,2,4,3)(5,7,8,6)(9,11,14,26,16,27,15,34)(10,17,20,23,22,36,21,39)(12,29,30,24,31,13,25,33)(18,19,35,32,28,38,40,37);(9,12,15,25,16,31,14,30)(10,18,21,40,22,28,20,35)(11,24,34,29,27,33,26,13)(17,32,39,19,36,37,23,38);(1,3,4,2)(9,13,14,33,16,29,15,24)(10,19,20,32,22,38,21,37)(11,25,26,12,27,30,34,31)(17,35,23,28,36,40,39,18)

Polycyclic presentation:

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

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, 276398);