#114629

order	512
nilpotency class	3
isoclinism	isoclinic class of #114623
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

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	#60951	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

815 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)(17,18,22,27,23,26,21,35)(19,28,30,39,31,24,37,40)(20,32,34,25,33,43,36,42)(29,44,47,38,45,48,46,41);(2,7)(3,11)(4,5)(9,10)(12,13)(14,15)(17,19)(18,24)(20,29)(21,30)(22,37)(23,31)(25,41)(26,28)(27,39)(32,44)(33,45)(34,46)(35,40)(36,47)(38,42)(43,48);(1,3)(2,8)(4,13)(5,11)(6,12)(7,14)(9,15)(10,16)(17,20)(18,25)(19,29)(21,36)(22,34)(23,33)(24,38)(26,42)(27,43)(28,41)(30,47)(31,45)(32,35)(37,46)(39,44)(40,48)

Polycyclic presentation:

2^9|a^2=g,d^2=i,e^2=h,f^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,f^b=fi,g^b=ghi,d^c=dhi,e^c=eh,f^c=fi

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, 21825);