#7166237

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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1664 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)(23,32)(24,30)(25,28)(26,31)(27,42)(29,35)(33,49)(34,40)(36,48)(37,50)(38,46)(39,41)(43,47)(44,51)(45,52);(7,13)(8,16)(10,17)(11,15)(21,23,28,36)(22,29,25,40)(24,37,39,33)(26,35,27,34)(30,45,41,44)(31,48,42,32)(38,51,43,52)(46,50,47,49);(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)(26,43,27,38)(29,44,40,45)(31,47,42,46)(32,49,48,50)(34,52,35,51);(7,13)(10,17)(14,19)(18,20)(21,25,28,22)(23,32,36,48)(24,30,39,41)(26,42,27,31)(29,34,40,35)(33,50,37,49)(38,46,43,47)(44,51,45,52);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,26,28,27)(22,31,25,42)(23,34,36,35)(24,38,39,43)(29,32,40,48)(30,46,41,47)(33,51,37,52)(44,50,45,49)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,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, 7073433);