#180925

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

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	C₂²×C₄⋊C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1392 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)(12,18)(13,14)(16,17)(19,20)(21,22,24,27,25,28,31,34)(23,26,30,33,29,32,35,36);(2,4)(5,6,9,13,10,14,8,11)(7,12,17,20,15,19,16,18)(21,23)(22,26)(24,30)(25,29)(27,33)(28,32)(31,35)(34,36);(5,7)(6,12)(8,16)(9,17)(10,15)(11,18)(13,20)(14,19)(23,29)(26,32)(30,35)(33,36);(7,15)(12,19)(16,17)(18,20)

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=h,f^2=i,g^2=h,b^a=be,e^a=eh,g^a=gh,c^b=ci,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, 88121);