#369200

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
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

1840 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)(13,30)(14,21)(15,20)(16,22)(17,26)(18,29)(19,24)(25,39)(27,40)(31,38)(32,35)(33,37)(34,36);(1,2,4,3)(5,7,8,6)(9,11,14,26,16,27,15,33)(10,17,20,23,22,37,21,40)(12,13,25,24,29,32,34,31)(18,38,39,35,28,19,36,30);(9,12,15,34,16,29,14,25)(10,18,21,36,22,28,20,39)(11,24,33,13,27,31,26,32)(17,35,40,38,37,30,23,19);(1,3,4,2)(9,13,15,31,16,32,14,24)(10,19,21,35,22,38,20,30)(11,25,33,12,27,34,26,29)(17,36,40,28,37,39,23,18)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fg,d^2=ei,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,f^c=fi,g^c=gi

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, 276396);