#182506

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
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

1048 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)(5,6,11,14)(7,15,17,19)(8,12,18,16)(9,13,10,20)(21,22,25,29,26,24,28,33)(23,27,32,35,31,30,34,36);(2,4)(5,7,10,16,11,17,9,12)(6,8,13,15,14,18,20,19)(21,23)(22,27)(24,30)(25,32)(26,31)(28,34)(29,35)(33,36);(5,8)(6,12)(7,13)(9,19)(10,15)(11,18)(14,16)(17,20)(21,24,25,33,26,22,28,29)(23,30,32,36,31,27,34,35);(5,9,11,10)(6,13,14,20)(7,12,17,16)(8,15,18,19)(23,31)(27,30)(32,34)(35,36)

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,e^a=eh,g^a=gh,d^b=di,d^c=dh

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 89702);