#6436592

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2938 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)(7,10)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,22,26,30)(23,31,32,44)(24,35,36,28)(25,38,40,29)(27,39,41,34)(33,47,46,45)(37,43,51,50)(42,48,49,52);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23)(22,27)(24,33)(25,39)(26,32)(28,42)(29,44)(30,41)(31,38)(34,40)(35,49)(36,46)(37,52)(43,47)(45,50)(48,51);(2,4)(7,13)(10,16)(14,19)(17,20)(23,32)(24,36)(27,41)(28,35)(31,44)(34,39)(37,51)(43,50);(6,11)(10,16)(12,18)(17,20)(21,24)(22,28)(23,33)(25,37)(26,36)(27,42)(29,43)(30,35)(31,45)(32,46)(34,48)(38,50)(39,52)(40,51)(41,49)(44,47);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(24,36)(28,35)(33,46)(37,51)(42,49)(43,50)(45,47)(48,52)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343788);