#120702

order	512
nilpotency class	3
isoclinism	isoclinic class of #8603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#117
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#382
prime radical	C₂⁴	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

526 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:

(5,6,7,9,8,10,11,12)(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)(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)(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,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,g^b=gh,g^c=gh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27898);