#454360

order	512
nilpotency class	3
isoclinism	isoclinic class of #453972
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2017	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1833 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=i,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,g^b=gi,d^c=de,f^c=fi,g^c=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, 361556);