#206984

order	512
nilpotency class	3
isoclinism	isoclinic class of #161742
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	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

926 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)(4,5)(6,8)(9,10)(11,18)(12,16)(13,22)(14,17)(15,21)(19,24)(20,23)(25,26)(27,30,33,37)(28,35)(29,32)(31,36)(34,40,39,38);(2,5)(6,7)(9,11,13,21,14,19,12,20)(10,15,16,18,17,23,22,24)(25,27)(26,30,32,37)(28,34)(29,33)(31,38,35,40)(36,39);(1,3,4,8)(2,6,5,7)(11,19)(15,23)(18,24)(20,21)(27,33)(30,37)(34,39)(38,40);(9,12,14,13)(10,16,17,22)(11,20,19,21)(15,18,23,24)(25,28,29,36)(26,31,32,35)(27,34,33,39)(30,38,37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,d^2=h,e^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ch,e^b=ei,g^b=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 114180);