#387958

order	512
nilpotency class	3
isoclinism	isoclinic class of #213808
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1182 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)(3,5)(4,7)(6,8)(9,10)(11,19,22,15)(12,23,25,32)(13,17)(14,18)(16,24,33,27)(20,36)(21,34,26,30)(28,35)(29,37)(31,39)(38,40);(1,3)(2,4)(5,8)(6,7)(9,11)(10,15)(12,20)(13,26)(14,22)(16,31)(17,34)(18,19)(21,28)(23,40)(24,39)(25,38)(27,37)(29,33)(30,35)(32,36);(1,4,6,5)(2,7,8,3)(9,12,14,25)(10,16,18,33)(11,20,22,38)(13,27,28,24)(15,29,19,31)(17,32,35,23)(21,39,26,37)(30,36,34,40);(1,5,6,4)(2,7,8,3)(10,17)(13,28)(15,30)(16,32)(18,35)(19,34)(21,26)(23,33)(24,27)(29,36)(31,40)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 295154);