#108601

order	512
nilpotency class	3
isoclinism	isoclinic class of #8803
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#383
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#126
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

460 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)(7,15,18,19)(8,11)(9,13)(10,14)(12,17,20,16)(21,22)(23,31,33,27)(24,28)(25,30)(26,29)(32,36,34,35);(6,11,14,13)(12,19,20,15)(21,23,25,34,26,33,24,32)(22,27,30,36,29,31,28,35);(2,4)(5,7,8,16,10,18,9,17)(6,12,13,15,14,20,11,19)(22,28,29,30)(27,35,31,36)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=hi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=ghi,f^b=fi,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, 15797);