#456519

order	512
nilpotency class	3
isoclinism	isoclinic class of #455962
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 363715);