#420527

order	512
nilpotency class	3
isoclinism	isoclinic class of #417661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1644 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327723);