#7182789

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1698 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,c^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7089985);