#224871

order	512
nilpotency class	3
isoclinism	isoclinic class of #165149
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

839 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,c^2=eh,d^2=ehi,f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ch,d^b=de,f^c=fi,g^c=ghi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 132067);