#110699

order	512
nilpotency class	3
isoclinism	isoclinic class of #9369
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1437
prime radical	C₂⁴	#126
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

555 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17895);