#348216

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

934 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)(5,6,9,12)(7,10,14,17)(8,11,15,18)(13,19,20,16)(22,26)(24,31)(27,35)(28,32)(30,33)(34,36);(1,2,3,4)(23,29)(27,36)(30,33)(34,35);(5,7)(6,10)(8,13)(9,14)(11,16)(12,17)(15,20)(18,19)(21,22,25,28)(23,27,29,36)(24,32,31,26)(30,34,33,35);(5,8,9,15)(6,11,12,18)(7,13,14,20)(10,16,17,19)(21,23,25,29)(22,27,28,36)(24,33,31,30)(26,34,32,35)

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=h,d^2=i,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, 255412);