#211390

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

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

674 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)(6,10)(8,15)(11,16)(12,17)(14,19)(18,20)(21,22,24,28)(23,30,31,26)(25,27,33,29)(32,35,36,34);(5,6,8,10,9,12,15,17)(7,11,14,16,13,18,19,20)(21,23)(22,26)(24,31)(25,32)(27,34)(28,30)(29,35)(33,36);(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(22,27)(25,33)(26,34)(28,29)(30,35)(32,36);(1,3,4,2)(6,12)(7,13)(10,17)(14,19)(23,31)(26,30)(32,36)(34,35)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=gi,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,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, 118586);