#205394

order	512
nilpotency class	3
isoclinism	isoclinic class of #19504
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1024 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=h,e^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112590);