#6569723

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3297	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2082 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6476919);