#205220

order	512
nilpotency class	3
isoclinism	isoclinic class of #19186
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 52 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112416);