#6687764

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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1731 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fg,d^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,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, 6594960);