#120678

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

498 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)(15,23,27,19)(16,28,30,43)(17,22)(18,21)(20,29,34,38)(24,31)(25,39)(26,36,37,33)(32,40)(35,44)(41,42);(1,2)(3,4)(5,6,8,11,10,12,9,7)(13,15,24,37,18,27,17,26)(14,19,31,36,21,23,22,33)(16,25,38,44,30,41,29,40)(20,32,43,39,34,35,28,42);(2,4)(5,7,8,6,10,11,9,12)(13,16)(14,20,21,34)(15,25)(17,29)(18,30)(19,32,23,35)(22,28,31,43)(24,38)(26,40)(27,41)(33,42,36,39)(37,44)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,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, 27874);