#239139

order	512
nilpotency class	3
isoclinism	isoclinic class of #233136
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1014 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,f^b=fi,g^b=gi,d^c=di,f^c=fhi,g^c=ghi

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, 146335);