#6529674

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500186
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2719 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)(21,22,25,29,26,30,39,45)(23,31,33,41,34,44,38,27)(24,35,37,51,36,28,42,43)(32,47,49,40,48,46,52,50);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,26,34)(22,27,30,41)(24,32,36,48)(25,38,39,33)(28,40,35,50)(29,44,45,31)(37,52,42,49)(43,47,51,46);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,24)(22,28)(23,32)(25,37)(26,36)(27,40)(29,43)(30,35)(31,46)(33,49)(34,48)(38,52)(39,42)(41,50)(44,47)(45,51);(6,12)(10,13)(11,20)(18,19)(24,36)(28,35)(32,48)(37,42)(40,50)(43,51)(46,47)(49,52)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6436870);