#213618

order	512
nilpotency class	3
isoclinism	isoclinic class of #163012
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1122 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ceh,f^c=fi,g^c=ghi

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, 120814);