#7167110

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1748 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,28,34)(23,35,40,46)(24,30,25,31)(26,33,27,32)(29,38,47,39)(36,45,42,52)(37,44,51,49)(41,50,43,48);(7,13)(8,16)(10,17)(11,15)(21,23,26,38,28,40,27,39)(22,29,32,46,34,47,33,35)(24,36,43,51,25,42,41,37)(30,44,50,52,31,49,48,45);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,24,28,25)(22,30,34,31)(23,36,40,42)(26,43,27,41)(29,44,47,49)(32,50,33,48)(35,45,46,52)(37,38,51,39);(7,13)(10,17)(14,19)(18,20)(21,25,28,24)(22,31,34,30)(23,37,40,51)(26,43,27,41)(29,45,47,52)(32,50,33,48)(35,44,46,49)(36,38,42,39);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,26,28,27)(22,32,34,33)(23,38,40,39)(24,41,25,43)(29,46,47,35)(30,48,31,50)(36,37,42,51)(44,45,49,52)

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei,f^d=fi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7074306);