#269504

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	#59034	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1016 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=h,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, 176700);