#186052

order	512
nilpotency class	3
isoclinism	isoclinic class of #18414
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#538	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,d^c=dhi,e^c=eh,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, 93248);