#6687358

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682596
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1770 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)(9,13)(10,14)(11,17)(16,20)(18,19)(21,22,26,30)(23,27,32,40)(24,34,35,49)(25,37,38,29)(28,42,43,36)(31,44,45,52)(33,48,47,41)(39,50,51,46);(1,2)(3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23)(22,27)(24,31)(25,33)(26,32)(28,39)(29,41)(30,40)(34,44)(35,45)(36,46)(37,48)(38,47)(42,50)(43,51)(49,52);(2,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,24,25,36,26,35,38,42)(22,28,29,34,30,43,37,49)(23,31,33,46,32,45,47,50)(27,39,41,44,40,51,48,52);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(23,32)(24,35)(27,40)(28,43)(33,47)(34,49)(36,42)(41,48);(2,4)(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(24,35)(28,43)(31,45)(34,49)(36,42)(39,51)(44,52)(46,50)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=f,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,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6594554);