#114419

order	512
nilpotency class	3
isoclinism	isoclinic class of #114414
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	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

761 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=di,f^b=fi,g^b=ghi,d^c=dh,f^c=fi,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, 21615);