#348223

order	512
nilpotency class	3
isoclinism	isoclinic class of #346398
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

982 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=h,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,d^b=dh,f^c=fh,g^c=gi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255419);