#186773

order	512
nilpotency class	3
isoclinism	isoclinic class of #18930
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×#3552	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2204 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)(5,6)(7,15)(8,11)(9,14)(10,13)(12,18)(16,19)(17,20)(21,22,26,30,27,23,25,29)(24,28,34,36,32,31,33,35);(2,4)(6,11)(8,9)(12,19)(13,14)(15,20)(17,18)(21,23,27,22)(24,31,32,28)(25,29,26,30)(33,35,34,36);(5,7)(6,12)(8,18)(9,17)(10,16)(11,19)(13,20)(14,15)(21,24)(22,28)(23,31)(25,33)(26,34)(27,32)(29,35)(30,36);(6,13)(7,16)(11,14)(17,18)(24,32)(28,31)(33,34)(35,36)

Polycyclic presentation:

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

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, 93969);