#187647

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

2060 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)(6,11,13,16)(12,19,20,18)(21,22,26,30,27,24,25,29)(23,31,34,28,35,36,33,32);(5,6)(7,12)(8,11)(9,16)(10,13)(14,18)(15,20)(17,19)(21,23,27,35)(22,28,24,32)(25,33,26,34)(29,31,30,36);(2,4)(5,7)(6,12)(8,14)(9,17)(10,15)(11,18)(13,20)(16,19)(21,24)(22,27)(23,32)(25,29)(26,30)(28,35)(31,33)(34,36);(5,8,10,9)(6,11,13,16)(7,14,15,17)(12,18,20,19)(21,25,27,26)(22,29,24,30)(23,33,35,34)(28,31,32,36)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^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, 94843);