#216189

order	512
nilpotency class	3
isoclinism	isoclinic class of #195990
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

897 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)(3,7)(9,10)(11,14)(12,19,20,15)(13,16)(17,23,24,21)(18,22)(25,26,29,34,30,32,28,33)(27,31,36,40,37,38,35,39);(1,2)(3,6)(4,5)(7,8)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23)(25,27,30,37)(26,31,32,38)(28,35,29,36)(33,39,34,40);(2,5)(6,8)(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23)(26,32)(28,29)(31,38)(35,36);(1,3,4,7)(2,6,5,8)(25,28,30,29)(26,33,32,34)(27,35,37,36)(31,39,38,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=ei,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^c=di,g^c=gh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 123385);