#121729

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

568 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=deh,d^2=h,e^2=i,g^2=h,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,d^b=dh,e^b=eh,g^b=gh,d^c=di,e^c=ei,g^c=gi

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, 28925);