#187643

order	512
nilpotency class	3
isoclinism	isoclinic class of #19014
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2066 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)(6,10,12,15)(11,19,20,18)(21,22,26,29,27,23,25,30)(24,32,34,28,33,36,35,31);(5,6)(7,11)(8,15)(9,12)(10,17)(13,18)(14,20)(16,19)(21,23,27,22)(24,31,33,28)(25,29,26,30)(32,35,36,34);(2,4)(5,7,9,14)(6,11,12,20)(8,16,17,13)(10,18,15,19)(21,24,27,33)(22,28,23,31)(25,34,26,35)(29,32,30,36);(24,33)(28,31)(32,36)(34,35)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,e^b=ei,f^b=fhi,g^b=ghi,d^c=dh,e^c=ehi,f^c=fh,g^c=gh

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, 94839);