#7125451

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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1734 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)(7,10)(8,11)(9,12)(13,15)(14,18)(16,19)(17,20)(21,22,26,30)(23,31,35,50)(24,28,37,45)(25,39,32,29)(27,41,44,36)(33,48,52,43)(34,49,51,42)(38,47,40,46);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,32,41,26,35,25,36)(22,27,39,50,30,44,29,31)(24,33,40,34,37,52,38,51)(28,42,47,43,45,49,46,48);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24)(22,28)(23,33)(25,38)(26,37)(27,42)(29,46)(30,45)(31,48)(32,40)(34,41)(35,52)(36,51)(39,47)(43,50)(44,49);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,24)(22,28)(23,34)(25,40)(26,37)(27,43)(29,47)(30,45)(31,49)(32,38)(33,41)(35,51)(36,52)(39,46)(42,50)(44,48);(6,12)(10,13)(11,20)(18,19)(23,35)(24,37)(27,44)(28,45)(31,50)(36,41)(38,40)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=h,d^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,e^d=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, 7032647);