#204021

order	512
nilpotency class	3
isoclinism	isoclinic class of #19456
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

544 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=eh,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 111217);