#6815524

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1739 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgh,d^2=f,e^2=hi,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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6722720);