#6807353

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1808 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fhi,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,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, 6714549);