#7180952

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1691 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)(8,16)(11,17)(15,20)(18,19)(21,22,28,46,27,32,26,33)(23,34,41,29,37,50,38,48)(24,39,44,47,36,30,42,52)(25,43,40,49,45,31,35,51);(5,6,9,12)(7,13,14,10)(8,17,16,11)(15,18,20,19)(21,23)(22,29)(24,40)(25,44)(26,41)(27,37)(28,38)(30,43)(31,39)(32,48)(33,50)(34,46)(35,36)(42,45)(47,49)(51,52);(2,4)(5,7)(6,10)(8,15)(9,14)(11,18)(12,13)(16,20)(17,19)(21,24)(22,30)(23,35)(25,41)(26,42)(27,36)(28,44)(29,31)(32,39)(33,47)(34,49)(37,40)(38,45)(43,48)(46,52)(50,51);(7,14)(10,13)(15,20)(18,19)(21,25,26,35,27,45,28,40)(22,31,33,49,32,43,46,51)(23,36,38,44,37,24,41,42)(29,47,34,39,48,52,50,30);(5,8,9,16)(6,11,12,17)(7,15,14,20)(10,18,13,19)(22,32)(23,37)(24,36)(31,43)(33,46)(38,41)(42,44)(49,51)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,d^2=f,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7088148);