#108065

order	512
nilpotency class	3
isoclinism	isoclinic class of #9279
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

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	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1044 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2)(3,4)(5,6,8,10,9,12,7,11)(13,14,18,22,19,23,17,33)(15,24,28,35,27,39,26,20)(16,29,31,37,32,21,30,36)(25,38,42,34,41,44,40,43);(2,4)(6,10)(7,8)(11,12)(13,15)(14,20,23,35)(16,25)(17,26)(18,28)(19,27)(21,34,29,43)(22,24,33,39)(30,40)(31,42)(32,41)(36,44,37,38);(6,10)(7,8)(11,12)(13,16,19,32)(14,21,23,29)(15,25,27,41)(17,31,18,30)(20,34,35,43)(22,37,33,36)(24,38,39,44)(26,42,28,40)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=hi,e^2=i,f^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, 15261);