#197988

order	512
nilpotency class	3
isoclinism	isoclinic class of #19552
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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

1034 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105184);