#182585

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	C₂×#121	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1028 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)(6,11)(8,9)(12,19)(13,14)(15,20)(17,18)(21,22,26,29,25,28,34,36)(23,30,32,33,31,27,35,24);(2,4)(5,6,9,14,10,13,8,11)(7,15,18,19,16,12,17,20)(21,23)(22,27)(24,29)(25,31)(26,32)(28,30)(33,36)(34,35);(5,7)(6,12)(8,17)(9,18)(10,16)(11,19)(13,15)(14,20)(21,24,26,30,25,33,34,27)(22,23,29,32,28,31,36,35);(6,13)(7,16)(11,14)(17,18)(23,31)(24,33)(27,30)(32,35)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,e^2=h,f^2=i,g^2=h,b^a=be,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, 89781);