#220106

order	512
nilpotency class	3
isoclinism	isoclinic class of #164512
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

996 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)(5,6,8,12,9,11,15,18)(7,10,13,17,14,16,19,20)(21,22)(23,30)(24,32)(25,29)(26,27)(28,31)(33,36)(34,35);(1,2)(3,4)(5,7,9,14)(6,10,11,16)(8,13,15,19)(12,17,18,20)(21,23,25,26)(22,27,29,30)(24,31,33,34)(28,32,35,36);(6,11)(8,15)(10,16)(13,19)(21,24,25,33)(22,28,29,35)(23,31,26,34)(27,32,30,36);(2,4)(22,29)(27,30)(28,35)(32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,f^a=fh,d^b=dei,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, 127302);