#7183785

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1649 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:

(1,2,4,3)(8,16)(11,17)(15,20)(18,19)(21,22,25,30,26,31,38,28)(23,27,33,41,34,42,49,39)(24,35,37,43,36,29,44,45)(32,46,48,50,47,40,51,52);(2,3)(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23,26,34)(22,27,31,42)(24,32,36,47)(25,33,38,49)(28,39,30,41)(29,40,35,46)(37,48,44,51)(43,50,45,52);(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(22,28)(25,38)(27,39)(29,43)(30,31)(33,49)(35,45)(37,44)(40,50)(41,42)(46,52)(48,51);(1,3,4,2)(7,14)(10,13)(15,20)(18,19)(21,24)(22,29)(23,32)(25,37)(26,36)(27,40)(28,43)(30,45)(31,35)(33,48)(34,47)(38,44)(39,50)(41,52)(42,46)(49,51);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(24,36)(29,35)(32,47)(37,44)(40,46)(43,45)(48,51)(50,52)

Polycyclic presentation:

2^9|a^2=fg,b^2=hi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

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, 7090981);