#194590

order	512
nilpotency class	3
isoclinism	isoclinic class of #19086
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	C₂×#121	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1106 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)(5,6,11,14,10,7,9,13)(8,12,16,19,18,15,17,20)(21,22,26,32,27,30,25,31)(23,33,37,28,35,43,36,44)(24,38,40,29,41,46,39,45)(34,42,50,51,48,52,49,47);(1,2,3,4)(5,7,10,6)(8,15,18,12)(9,13,11,14)(16,19,17,20)(21,23)(22,28)(24,34)(25,37)(26,36)(27,35)(29,42)(30,44)(31,43)(32,33)(38,51)(39,50)(40,49)(41,48)(45,52)(46,47);(5,8)(6,12)(7,15)(9,16)(10,18)(11,17)(13,19)(14,20)(21,24)(22,29)(23,34)(25,39)(26,40)(27,41)(28,42)(30,45)(31,38)(32,46)(33,47)(35,48)(36,49)(37,50)(43,51)(44,52);(5,9,10,11)(6,13,7,14)(8,16,18,17)(12,19,15,20)(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=egi,d^2=i,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dhi,f^a=fh,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,f^c=fhi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101786);