#233452

order	512
nilpotency class	3
isoclinism	isoclinic class of #233431
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1382 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)(7,17)(8,19)(9,16)(10,14)(11,15)(12,13)(18,20)(21,22,26,29)(23,31,32,35)(24,30,25,28)(27,34,36,33);(2,4)(5,7)(6,12)(8,14)(9,19)(10,18)(11,13)(15,20)(16,17)(21,23)(22,27)(24,34)(25,33)(26,32)(28,35)(29,36)(30,31);(5,8,10,12)(6,13,14,19)(7,16,18,15)(9,17,11,20)(22,28)(24,25)(27,35)(29,30)(31,36)(33,34);(6,14)(7,18)(13,19)(15,16)(22,29)(23,32)(28,30)(33,34)

Polycyclic presentation:

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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140648);