#121338

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#399
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

616 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=e,d^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^b=ei,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, 28534);