#113871

order	512
nilpotency class	3
isoclinism	isoclinic class of #113863
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	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

809 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:

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

Polycyclic presentation:

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

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, 21067);