#348240

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

974 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)(7,11)(8,12)(9,10)(13,19)(14,16)(15,17)(18,20)(21,22,25,24)(23,29,31,36)(26,33,34,28)(27,35,30,32);(1,2,3,4)(6,10)(11,16)(12,17)(18,19)(22,24)(27,30)(28,33)(29,36);(5,7)(6,11)(8,13)(9,14)(10,16)(12,18)(15,20)(17,19)(21,23,25,31)(22,27,24,30)(26,35,34,32)(28,36,33,29);(5,8,9,15)(6,12,10,17)(7,13,14,20)(11,18,16,19)(21,24,25,22)(23,30,31,27)(26,28,34,33)(29,35,36,32)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,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, 255436);