#378504

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

Polycyclic presentation:

2^9|a^2=h,c^2=ef,d^2=efh,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, 285700);