#7167084

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

1755 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)(21,22,26,30)(23,31,34,42)(24,28,37,45)(25,39,40,29)(27,33,43,35)(32,47,36,51)(38,52,48,46)(41,50,44,49);(7,13)(8,16)(10,17)(11,15)(21,23,25,35,26,34,40,33)(22,27,29,31,30,43,39,42)(24,36,38,50,37,32,48,49)(28,44,46,51,45,41,52,47);(2,4)(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,24,26,37)(22,28,30,45)(23,32,34,36)(25,38,40,48)(27,41,43,44)(29,46,39,52)(31,47,42,51)(33,50,35,49);(7,13)(10,17)(14,19)(18,20)(23,33)(24,37)(25,40)(27,42)(28,45)(29,39)(31,43)(32,49)(34,35)(36,50)(41,47)(44,51);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(23,34)(24,37)(27,43)(28,45)(31,42)(33,35)(38,48)(46,52)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^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, 7074280);