#205670

order	512
nilpotency class	3
isoclinism	isoclinic class of #205546
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	C₂⁴	#124
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

917 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)(3,9)(4,12)(5,7)(6,11)(8,13)(10,15)(14,16)(17,18,22,27,23,26,21,35)(19,24,30,40,29,38,36,39)(20,31,33,41,34,45,32,25)(28,42,44,37,43,48,46,47);(1,3,4,11,5,10,13,16)(2,6,8,9,7,14,12,15)(17,19,22,36,23,29,21,30)(18,24,35,40,26,38,27,39)(20,28,33,46,34,43,32,44)(25,37,45,48,41,47,31,42);(2,7)(6,14)(8,12)(9,15)(17,20,21,32,23,34,22,33)(18,25,27,31,26,41,35,45)(19,28,30,44,29,43,36,46)(24,37,39,42,38,47,40,48);(3,10)(6,14)(9,15)(11,16)(19,29)(24,38)(28,43)(30,36)(37,47)(39,40)(42,48)(44,46)

Polycyclic presentation:

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

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, 112866);