#269516

order	512
nilpotency class	3
isoclinism	isoclinic class of #269339
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

994 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,d^2=h,e^2=i,f^2=hi,b^a=be,c^a=cf,d^a=deh,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

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 176712);