#214440

order	512
nilpotency class	3
isoclinism	isoclinic class of #162948
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

979 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)(11,18)(12,16)(13,21)(14,17)(15,19)(20,23)(22,24)(27,33,34,35)(30,37,38,40);(1,3)(2,5)(4,8)(6,7)(25,26,29,31)(27,30,34,38)(28,32,36,39)(33,37,35,40);(3,8)(5,7)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21)(25,27)(26,30)(28,35)(29,34)(31,38)(32,40)(33,36)(37,39);(2,6)(5,7)(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,24,18)(26,31)(30,38)(32,39)(37,40)

Polycyclic presentation:

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

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, 121636);