#194536

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1030 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)(7,14,15,17)(11,18,20,19)(21,22,26,31,27,32,25,30)(23,33,36,28,37,43,35,44)(24,38,41,29,39,46,40,45)(34,42,50,51,48,52,49,47);(1,2,3,4)(5,6)(7,11)(8,12)(9,16)(10,13)(14,19)(15,20)(17,18)(21,23)(22,28)(24,34)(25,36)(26,35)(27,37)(29,42)(30,43)(31,33)(32,44)(38,51)(39,48)(40,50)(41,49)(45,52)(46,47);(5,7)(6,11)(8,14)(9,17)(10,15)(12,19)(13,20)(16,18)(21,24,27,39)(22,29,32,45)(23,34,37,48)(25,40,26,41)(28,42,44,52)(30,38,31,46)(33,47,43,51)(35,49,36,50);(5,8,10,9)(6,12,13,16)(7,14,15,17)(11,19,20,18)(24,39)(29,45)(34,48)(38,46)(40,41)(42,52)(47,51)(49,50)

Polycyclic presentation:

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

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, 101732);