#7166238

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×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1693 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)(21,22,28,33)(23,34,38,49)(24,30,25,40)(26,32,27,31)(29,43,46,37)(35,50,41,45)(36,44,52,51)(39,48,42,47);(7,13)(8,16)(10,17)(11,15)(21,23,28,38)(22,29,33,46)(24,35,25,41)(26,37,27,43)(30,44,40,51)(31,34,32,49)(36,39,52,42)(45,48,50,47);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,24,28,25)(22,30,33,40)(23,35,38,41)(26,42,27,39)(29,44,46,51)(31,48,32,47)(34,50,49,45)(36,43,52,37);(7,13)(10,17)(14,19)(18,20)(21,25,28,24)(22,30,33,40)(23,36,38,52)(26,42,27,39)(29,45,46,50)(31,47,32,48)(34,51,49,44)(35,43,41,37);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,26,28,27)(22,31,33,32)(23,37,38,43)(24,39,25,42)(29,34,46,49)(30,47,40,48)(35,52,41,36)(44,45,51,50)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,d^2=i,e^2=hi,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, 7073434);