#114062

order	512
nilpotency class	3
isoclinism	isoclinic class of #114053
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	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

780 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)(3,9,11,7)(4,5)(6,13)(8,14)(10,15,16,12)(17,18,22,28,23,27,21,26)(19,29,32,24,33,40,31,39)(20,34,35,25,37,42,36,43)(30,38,45,44,47,41,46,48);(1,3)(2,7)(4,9)(5,11)(6,15)(8,16)(10,13)(12,14)(17,19,23,33)(18,24,27,39)(20,30,37,47)(21,32,22,31)(25,41,43,38)(26,40,28,29)(34,48,42,44)(35,46,36,45);(1,4,5,2)(3,10,11,16)(6,8,14,13)(7,15,9,12)(17,20)(18,25)(19,30)(21,35)(22,36)(23,37)(24,38)(26,42)(27,43)(28,34)(29,44)(31,45)(32,46)(33,47)(39,41)(40,48)

Polycyclic presentation:

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

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, 21258);