#214457

order	512
nilpotency class	3
isoclinism	isoclinic class of #163298
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1198 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)(11,14)(12,19,21,15)(13,16)(17,24,20,22)(18,23)(25,26)(27,31)(28,35)(29,38)(30,33)(32,34)(36,40)(37,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,20)(13,18)(15,24)(16,23)(17,21)(19,22)(25,27,30,29)(26,31,33,38)(28,36,37,34)(32,35,40,39);(3,8)(5,7)(9,12,13,21)(10,15,16,19)(11,17,18,20)(14,22,23,24)(25,28,30,37)(26,32,33,40)(27,34,29,36)(31,39,38,35);(2,6)(5,7)(11,18)(14,23)(17,20)(22,24)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121653);