#182545

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

960 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)(8,12,17,20)(14,16,19,18)(21,22,26,29,25,28,34,36)(23,30,32,33,31,27,35,24);(2,4)(5,7,10,13,11,15,9,6)(8,14,16,20,17,19,18,12)(21,23)(22,27)(24,29)(25,31)(26,32)(28,30)(33,36)(34,35);(5,8)(6,12)(7,14)(9,18)(10,16)(11,17)(13,20)(15,19)(21,24,26,30,25,33,34,27)(22,23,29,32,28,31,36,35);(5,9,11,10)(6,7,13,15)(8,16,17,18)(12,19,20,14)(22,28)(23,31)(29,36)(32,35)

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,d^b=dhi,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, 89741);