#114564

order	512
nilpotency class	3
isoclinism	isoclinic class of #114551
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

805 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,d^2=i,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,f^c=fi,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21760);