#109969

order	512
nilpotency class	3
isoclinism	isoclinic class of #1595
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

586 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17165);