#223233

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

787 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)(9,10,13,16,14,17,12,21)(11,18,20,23,19,15,22,24)(25,26)(27,31)(28,35)(29,38)(30,33)(32,34)(36,40)(37,39);(1,2)(3,6)(4,5)(7,8)(11,19)(15,18)(20,22)(23,24)(25,27,30,29)(26,31,33,38)(28,36,37,34)(32,35,40,39);(1,3,4,7)(2,6,5,8)(9,11,14,19)(10,15,17,18)(12,20,13,22)(16,23,21,24)(25,28,30,37)(26,32,33,40)(27,34,29,36)(31,39,38,35);(2,5)(6,8)(9,12,14,13)(10,16,17,21)(11,20,19,22)(15,23,18,24)(25,27,30,29)(26,31,33,38)(28,34,37,36)(32,39,40,35)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=eh,d^2=h,f^2=i,g^2=hi,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, 130429);