#116323

order	512
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#385
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

657 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,e^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,e^a=ei,f^a=fi,c^b=cf,e^b=ehi,f^b=fhi,d^c=dh,e^c=ei,f^c=fi

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, 23519);