#223122

order	512
nilpotency class	3
isoclinism	isoclinic class of #165149
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

795 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dhi,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, 130318);