#376402

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2366
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

948 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 283598);