#121660

order	512
nilpotency class	3
isoclinism	isoclinic class of #10370
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#124
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#397
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

560 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)(7,10)(8,11)(9,12)(13,14)(15,23)(16,27)(17,30)(18,22)(19,25)(20,29)(21,32)(24,40)(26,39)(28,36)(31,37)(33,41)(34,44)(35,43)(38,42);(6,10)(7,8)(11,12)(13,15,17,25,18,26,32,44)(14,19,21,23,22,34,30,39)(16,24,31,43,29,42,28,41)(20,35,37,40,27,33,36,38);(6,11)(7,8)(10,12)(13,16,17,31,18,29,32,28)(14,20,21,37,22,27,30,36)(15,24,25,43,26,42,44,41)(19,33,23,38,34,35,39,40)

Polycyclic presentation:

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

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, 28856);