#115531

order	512
nilpotency class	3
isoclinism	isoclinic class of #115519
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	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#3219	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

842 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)(3,10,12,7)(4,6)(5,9)(8,13)(11,15,14,16)(17,18,22,26,23,27,21,28)(19,29,31,24,32,41,38,40)(20,33,35,43,37,25,36,44)(30,45,34,39,47,42,46,48);(2,6)(4,13)(7,16)(8,9)(10,15)(11,14)(17,19)(18,24)(20,34)(21,38)(22,31)(23,32)(25,42)(26,41)(27,40)(28,29)(30,36)(33,45)(35,47)(37,46)(39,43)(44,48);(1,3)(2,7)(4,14)(5,12)(6,15)(8,16)(9,10)(11,13)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,39)(26,43)(27,33)(28,44)(29,42)(31,46)(32,47)(34,38)(40,48)(41,45)

Polycyclic presentation:

2^9|a^2=g,d^2=h,f^2=hi,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=dh,f^b=fhi,d^c=di,f^c=fhi,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, 22727);