#366631

order	512
nilpotency class	3
isoclinism	isoclinic class of #25798
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2223 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)(5,6,9,15,11,14,10,16)(7,17,20,18,21,12,25,28)(8,22,23,30,24,13,26,29)(19,27,33,32,35,31,34,36);(1,2)(3,4)(5,7,9,25,11,21,10,20)(6,12,15,18,14,17,16,28)(8,19,26,33,24,35,23,34)(13,27,30,32,22,31,29,36);(2,4)(5,6)(7,18)(8,13)(9,16)(10,15)(11,14)(12,25)(17,20)(19,32)(21,28)(22,24)(23,30)(26,29)(27,33)(31,34)(35,36);(5,8)(6,13)(7,19)(9,26)(10,23)(11,24)(12,27)(14,22)(15,30)(16,29)(17,31)(18,32)(20,34)(21,35)(25,33)(28,36)

Polycyclic presentation:

2^9|a^2=h,b^2=fgh,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^b=fi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273827);