#6526768

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499814
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1704 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)(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22)(23,31)(24,28)(25,37)(26,29)(27,33)(30,39)(32,45)(34,44)(35,42)(36,49)(38,41)(40,47)(43,48)(46,52)(50,51);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,26,34)(22,27,29,41)(24,32,35,46)(25,38,39,33)(28,40,42,51)(30,44,37,31)(36,50,48,47)(43,52,49,45);(1,2,3,4)(7,13)(8,15)(10,16)(11,18)(21,24)(22,28)(23,32)(25,36)(26,35)(27,40)(29,42)(30,43)(31,45)(33,47)(34,46)(37,49)(38,50)(39,48)(41,51)(44,52);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(22,29)(24,35)(27,41)(30,37)(31,44)(32,46)(36,48)(47,50);(7,13)(10,16)(14,19)(17,20)(24,35)(28,42)(32,46)(36,48)(40,51)(43,49)(45,52)(47,50)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=h,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,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, 6433964);