#205566

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	#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

918 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,e^2=hi,f^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, 112762);