#205719

order	512
nilpotency class	3
isoclinism	isoclinic class of #205565
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

892 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(17,18,22,27,23,28,21,26)(19,29,32,24,33,39,31,40)(20,34,37,41,35,42,36,25)(30,44,47,43,45,38,46,48);(1,3,4,12,6,11,5,13)(2,7,14,16,9,15,8,10)(17,19)(18,24)(20,30)(21,32)(22,31)(23,33)(25,38)(26,39)(27,29)(28,40)(34,43)(35,45)(36,47)(37,46)(41,44)(42,48);(3,11)(7,15)(10,16)(12,13)(17,20,21,36,23,35,22,37)(18,25,27,34,28,41,26,42)(19,30,32,47,33,45,31,46)(24,38,29,43,40,44,39,48);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(20,35)(25,41)(30,45)(34,42)(36,37)(38,44)(43,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112915);