#269522

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

1003 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=h,d^2=hi,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, 176718);