#119040

order	512
nilpotency class	3
isoclinism	isoclinic class of #119038
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

489 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)(3,4,8,14)(5,9)(6,10)(7,12,13,15)(11,16)(17,18,22,28,23,26,21,27)(19,29,32,40,33,24,31,41)(20,34,36,43,38,44,37,25)(30,46,47,39,48,42,35,45);(1,3)(2,7)(4,11)(5,15)(6,8)(9,14)(10,13)(12,16)(17,19,23,33)(18,24,26,29)(20,35,38,47)(21,32,22,31)(25,42,43,46)(27,41,28,40)(30,36,48,37)(34,39,44,45);(1,4,5,13,6,14,16,7)(2,8,9,15,10,3,11,12)(17,20)(18,25,26,43)(19,30)(21,36)(22,37)(23,38)(24,39,29,45)(27,34,28,44)(31,47)(32,35)(33,48)(40,42,41,46)

Polycyclic presentation:

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

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, 26236);