#6526749

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499792
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1734 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)(25,38)(26,29)(27,34)(30,40)(32,47)(33,46)(35,49)(36,44)(37,52)(39,42)(41,48)(43,50)(45,51);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23)(22,27)(24,35)(25,39)(26,33)(28,43)(29,42)(30,46)(31,38)(32,36)(34,40)(37,48)(41,44)(45,47)(49,52)(50,51);(1,2,3,4)(7,13)(8,15)(10,16)(11,18)(21,24)(22,28)(23,32)(25,37)(26,36)(27,41)(29,44)(30,45)(31,47)(33,35)(34,48)(38,52)(39,50)(40,51)(42,43)(46,49);(5,8,9,15)(6,11,12,18)(7,14,13,19)(10,17,16,20)(22,29)(24,36)(27,42)(30,38)(31,46)(32,35)(37,51)(48,50);(7,13)(10,16)(14,19)(17,20)(23,33)(24,36)(27,42)(28,44)(31,46)(34,39)(37,51)(45,52)

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=hi,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=dh,e^c=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, 6433945);