#223073

order	512
nilpotency class	3
isoclinism	isoclinic class of #164187
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

886 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 130269);