#471443

order	512
nilpotency class	3
isoclinism	isoclinic class of #456670
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

1061 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)(5,6,11,17)(7,19,22,13)(8,25,28,40)(9,30,33,41)(10,18,12,16)(14,29,39,27)(15,31,42,32)(20,45,47,55)(21,50,52,61)(23,36,24,37)(26,57,60,38)(34,48,43,49)(35,54,44,53)(46,63,51,64)(56,58,62,59)(65,66,68,67);(5,7,10,23,11,22,12,24)(6,13,16,36,17,19,18,37)(8,20,27,48,28,47,29,49)(9,21,32,54,33,52,31,53)(14,34,25,55,39,43,40,45)(15,35,30,61,42,44,41,50)(26,46,59,67,60,51,58,66)(38,63,56,68,57,64,62,65);(2,4)(5,8,11,28)(6,14,17,39)(7,20,22,47)(9,26,33,60)(10,29,12,27)(13,34,19,43)(15,38,42,57)(16,40,18,25)(21,51,52,46)(23,49,24,48)(30,62,41,56)(31,59,32,58)(35,64,44,63)(36,45,37,55)(50,65,61,68)(53,66,54,67);(5,9)(6,15)(7,21)(8,26)(10,32)(11,33)(12,31)(13,35)(14,38)(16,30)(17,42)(18,41)(19,44)(20,46)(22,52)(23,54)(24,53)(25,56)(27,59)(28,60)(29,58)(34,63)(36,61)(37,50)(39,57)(40,62)(43,64)(45,65)(47,51)(48,67)(49,66)(55,68)

Polycyclic presentation:

2^9|a^2=i,b^2=egh,c^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fi,h^c=hi,f^d=fi,g^d=gi,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, 378639);