#223132

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	#2416	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

835 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)(9,10,13,16,12,15,20,23)(11,17,19,24,18,14,21,22)(25,26)(27,31)(28,36,37,40)(29,34)(30,33)(32,35,39,38);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,18)(13,19)(15,17)(16,22)(20,21)(23,24)(25,27,30,29)(26,31,33,34)(28,35,37,38)(32,36,39,40);(1,3,4,7)(2,6,5,8)(10,15)(11,18)(13,20)(22,24)(25,28)(26,32)(27,35)(29,38)(30,37)(31,36)(33,39)(34,40);(2,5)(6,8)(10,15)(11,18)(16,23)(19,21)(26,33)(31,34)(32,39)(36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=eh,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, 130328);