#120677

order	512
nilpotency class	3
isoclinism	isoclinic class of #99283
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#382
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

542 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=i,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,g^b=ghi,g^c=ghi

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, 27873);