#205603

order	512
nilpotency class	3
isoclinism	isoclinic class of #205553
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

891 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=f,d^2=h,e^2=hi,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=ei,g^b=gi,d^c=dh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112799);