#211687

order	512
nilpotency class	3
isoclinism	isoclinic class of #19552
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

662 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ehi,d^2=ghi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=dhi,d^c=dh,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 118883);