#110669

order	512
nilpotency class	3
isoclinism	isoclinic class of #9307
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

628 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di

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, 17865);