#7124580

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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1661 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)(21,22,28,34)(23,35,40,46)(24,30,25,31)(26,33,27,32)(29,38,47,39)(36,45,41,52)(37,44,51,48)(42,50,43,49);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,28,40)(22,29,34,47)(24,41,25,36)(26,39,27,38)(30,48,31,44)(32,35,33,46)(37,43,51,42)(45,50,52,49);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24,28,25)(22,30,34,31)(23,36,40,41)(26,43,27,42)(29,44,47,48)(32,50,33,49)(35,45,46,52)(37,38,51,39);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,25,28,24)(22,31,34,30)(23,37,40,51)(26,43,27,42)(29,45,47,52)(32,50,33,49)(35,44,46,48)(36,38,41,39);(6,12)(10,13)(11,20)(18,19)(21,26,28,27)(22,32,34,33)(23,38,40,39)(24,42,25,43)(29,46,47,35)(30,49,31,50)(36,37,41,51)(44,45,48,52)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,f^b=fi,d^c=dhi,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, 7031776);