#211674

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, prime radical	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=i,d^2=gi,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, 118870);