#183231

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=fh,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,d^a=di,e^a=eh,g^a=gh,c^b=ch,d^b=dhi,d^c=dhi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90427);