#189838

order	512
nilpotency class	3
isoclinism	isoclinic class of #161128
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1186 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,20,22,16)(13,17)(14,18)(19,24,21,23)(25,26)(27,30)(28,31)(29,32)(33,36)(34,37)(35,38)(39,40);(2,5)(6,8)(9,11,14,13)(10,15,18,17)(12,19,22,21)(16,23,20,24)(25,27,29,35)(26,30,32,38)(28,34,36,40)(31,37,33,39);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20)(26,31)(28,36)(30,37)(32,33)(34,40)(38,39);(9,13,14,11)(10,17,18,15)(12,21,22,19)(16,24,20,23)(26,32)(30,38)(31,33)(37,39)

Polycyclic presentation:

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

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, 97034);