#463541

order	512
nilpotency class	3
isoclinism	isoclinic class of #463540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16476
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

872 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=i,c^2=ef,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,g^d=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 370737);