#107468

order	512
nilpotency class	3
isoclinism	isoclinic class of #9331
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

701 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,10,7,11,6,9)(3,12,13,16,14,15,8,4)(17,18)(19,29)(20,33)(21,27)(22,26)(23,28)(24,31)(25,35)(30,38)(32,42)(34,41)(36,40)(37,39)(43,44)(45,48)(46,47);(1,3,6,8,7,14,5,13)(2,4,9,15,11,16,10,12)(17,19)(18,24,28,39)(20,30)(21,31)(22,37)(23,32)(25,38,41,48)(26,29,27,42)(33,43,40,47)(34,44)(35,46)(36,45);(1,4,7,16)(2,8,11,13)(3,9,14,10)(5,12,6,15)(17,20,23,36)(18,25,28,41)(19,30,32,45)(21,35,22,34)(24,38,39,48)(26,33,27,40)(29,43,42,47)(31,46,37,44)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,d^b=di,e^b=ehi,d^c=dhi,e^c=eh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14664);