#268555

order	512
nilpotency class	3
isoclinism	isoclinic class of #268491
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	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

974 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=h,e^2=i,f^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,e^c=ehi,f^c=fhi,e^d=ei,f^d=fhi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 175751);