#233204

order	512
nilpotency class	3
isoclinism	isoclinic class of #233184
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1246 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)(7,15)(8,11)(9,13)(10,14)(12,18)(16,20)(17,19)(21,22,25,28)(23,30,32,35)(24,27,34,29)(26,31,36,33);(2,4)(6,11)(7,16)(8,10)(13,14)(17,18)(19,20)(21,23)(22,26)(24,31)(25,32)(27,30)(28,36)(29,35)(33,34);(5,7,9,18)(6,12,13,15)(8,16,10,17)(11,19,14,20)(22,27)(23,31)(24,34)(28,29)(30,35)(32,33);(6,13)(11,14)(12,15)(19,20)(22,28)(23,32)(27,29)(31,33)

Polycyclic presentation:

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

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, 140400);