#385898

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	D₄×C₂²⋊C₄
prime radical	#60450	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1413 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)(3,4)(5,7)(6,8)(9,10,13,19,15,20,14,18)(11,16,22,32,23,33,29,31)(12,24,27,36,28,35,26,17)(21,34,25,39,38,40,37,30);(1,3)(2,6)(4,7)(5,8)(9,11)(10,16)(12,25)(13,22)(14,29)(15,23)(17,34)(18,31)(19,32)(20,33)(21,26)(24,39)(27,38)(28,37)(30,35)(36,40);(1,4,5,6)(2,3,7,8)(9,12,14,26,15,28,13,27)(10,17,18,35,20,36,19,24)(11,21,22,25,23,38,29,37)(16,30,32,34,33,39,31,40);(2,7)(3,8)(9,10)(11,16)(12,17)(13,19)(14,18)(15,20)(21,30)(22,32)(23,33)(24,27)(25,34)(26,35)(28,36)(29,31)(37,40)(38,39)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293094);