#121360

order	512
nilpotency class	3
isoclinism	isoclinic class of #9914
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#385
prime radical	C₂⁴	C₂×C₄⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

598 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=dh,c^2=e,d^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,f^b=fi,d^c=di

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, 28556);