#152378

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₁₆	C₂³
commutator, socle, prime radical	C₂³	C₂²×C₁₆
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

214 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 76
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 76 points:

(2,3)(5,6)(7,10)(8,9)(11,12)(13,14,16,21,17,22,29,39,18,23,30,40,32,42,51,61,19,24,31,41,33,43,52,62,34,44,53,63,54,64,69,74)(15,20,26,36,27,37,48,58,28,38,49,59,50,60,68,73,25,35,45,55,46,56,65,70,47,57,66,71,67,72,75,76);(1,2,4,3)(6,9)(10,12)(13,15)(14,20)(16,26)(17,27)(18,28)(19,25)(21,36)(22,37)(23,38)(24,35)(29,48)(30,49)(31,45)(32,50)(33,46)(34,47)(39,58)(40,59)(41,55)(42,60)(43,56)(44,57)(51,68)(52,65)(53,66)(54,67)(61,73)(62,70)(63,71)(64,72)(69,75)(74,76);(1,3,4,2)(5,7,8,11)(6,10,9,12)(15,25)(20,35)(26,45)(27,46)(28,47)(36,55)(37,56)(38,57)(48,65)(49,66)(50,67)(58,70)(59,71)(60,72)(68,75)(73,76)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=d,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=ci

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59574);