#7167125

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1726 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)(21,22,28,25)(23,32,36,34)(24,30,39,41)(26,31,27,42)(29,43,40,35)(33,49,37,52)(38,47,44,48)(45,50,46,51);(7,13)(8,16)(10,17)(11,15)(21,23,26,43,28,36,27,35)(22,29,42,34,25,40,31,32)(24,37,44,51,39,33,38,50)(30,46,47,49,41,45,48,52);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,24,28,39)(22,30,25,41)(23,33,36,37)(26,44,27,38)(29,45,40,46)(31,48,42,47)(32,49,34,52)(35,51,43,50);(7,13)(10,17)(14,19)(18,20)(21,25,28,22)(23,34,36,32)(24,30,39,41)(26,42,27,31)(29,35,40,43)(33,49,37,52)(38,47,44,48)(45,50,46,51);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,26,28,27)(22,31,25,42)(23,35,36,43)(24,38,39,44)(29,34,40,32)(30,47,41,48)(33,50,37,51)(45,49,46,52)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=i,d^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, 7074321);