#220111

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

945 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)(5,6,9,14,10,13,7,11)(8,16,15,20,17,12,18,19)(21,22)(23,31)(24,30)(25,27)(26,29)(28,32)(33,36)(34,35);(1,2)(3,4)(5,7,10,9)(6,11,13,14)(8,15,17,18)(12,19,16,20)(22,27)(28,35)(29,30)(31,36);(5,8)(6,12)(7,15)(9,18)(10,17)(11,19)(13,16)(14,20)(21,23,25,33)(22,28,27,35)(24,32,26,34)(29,31,30,36);(2,4)(5,7,10,9)(6,11,13,14)(8,15,17,18)(12,19,16,20)(21,24,25,26)(22,29,27,30)(23,32,33,34)(28,31,35,36)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,d^2=h,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, 127307);