#108050

order	512
nilpotency class	3
isoclinism	isoclinic class of #9279
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

884 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)(5,6,10,12)(7,9,14,18)(8,11)(13,15)(16,20)(17,19)(21,22)(23,26,30,35)(24,27)(25,28)(29,32)(31,33,36,34);(2,4)(5,7,9,12,10,14,18,6)(8,13,17,20,16,19,15,11)(21,23)(22,26,28,35)(24,31)(25,30)(27,33,29,34)(32,36);(5,8)(6,11)(7,13)(9,17)(10,16)(12,20)(14,19)(15,18)(22,27)(24,32)(26,33)(28,29)(31,36)(34,35)

Polycyclic presentation:

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

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, 15246);