#213589

order	512
nilpotency class	3
isoclinism	isoclinic class of #213564
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1202 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)(11,17,19,14)(12,15)(13,16)(18,23,24,21)(20,22)(25,26,29,33,30,34,28,32)(27,31,35,38,36,39,37,40);(1,3)(2,5)(4,8)(6,7)(12,20)(15,22)(18,24)(21,23)(25,27)(26,31)(28,37)(29,35)(30,36)(32,40)(33,38)(34,39);(3,8)(5,7)(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23);(2,6)(5,7)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28,30,29)(26,32,34,33)(27,35,36,37)(31,38,39,40)

Polycyclic presentation:

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

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, 120785);