#369832

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1128 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)(9,10,14,16,15,20,13,19)(11,21,24,17,25,32,22,30)(12,26,28,18,29,33,27,34)(23,31,36,35,38,39,37,40);(1,2,4,3)(5,7,8,6)(10,16)(11,22)(13,14)(17,30)(18,26)(19,20)(23,36)(24,25)(27,28)(31,39)(33,34)(37,38);(9,11,14,22,15,25,13,24)(10,17,16,21,20,30,19,32)(12,23,27,36,29,38,28,37)(18,31,26,35,34,39,33,40);(1,3,4,2)(9,12,15,29)(10,18,20,34)(11,23,25,38)(13,28,14,27)(16,26,19,33)(17,31,30,39)(21,35,32,40)(22,36,24,37)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 277028);