#111978

order	512
nilpotency class	3
isoclinism	isoclinic class of #111967
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

873 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19174);