#119315

order	512
nilpotency class	3
isoclinism	isoclinic class of #100003
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 52 points:

(5,6,8,12,9,11,17,19)(7,13,15,20,14,10,16,18)(21,22)(23,33)(24,29,40,48)(25,32)(26,30)(27,31)(28,35)(34,45,51,49)(36,43)(37,44)(38,42,50,52)(39,47,41,46);(1,2)(3,4)(5,7)(6,10)(8,16)(9,14)(11,13)(12,20)(15,17)(18,19)(21,23,27,36)(22,28,31,44)(24,38,40,50)(25,37,26,35)(29,45,48,49)(30,43,32,33)(34,39,51,41)(42,46,52,47);(2,4)(5,7)(6,10,11,13)(8,15)(9,14)(12,18,19,20)(16,17)(21,24,25,39,27,40,26,41)(22,29,30,46,31,48,32,47)(23,34,35,50,36,51,37,38)(28,42,33,49,44,52,43,45)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=i,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,e^c=eh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26511);