#220065

order	512
nilpotency class	3
isoclinism	isoclinic class of #165149
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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

1000 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^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, 127261);