#6674520

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#29606	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1837 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

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, 6581716);