#233513

order	512
nilpotency class	3
isoclinism	isoclinic class of #233501
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
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

1330 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)(5,6,10,12)(7,15,17,19)(8,14,9,13)(11,18,20,16)(21,22)(23,31)(24,30)(25,29)(26,28)(27,33)(32,36)(34,35);(2,4)(5,7)(6,11)(8,18)(9,16)(10,17)(12,20)(13,19)(14,15)(23,32)(24,25)(27,35)(29,30)(31,36)(33,34);(7,16)(8,9)(11,15)(13,14)(17,18)(19,20)(21,23,26,34)(22,27,28,36)(24,32,25,33)(29,35,30,31);(6,12)(7,17)(13,14)(16,18)(22,28)(27,36)(29,30)(31,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140709);