#423193

order	512
nilpotency class	3
isoclinism	isoclinic class of #421163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

1276 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ef,d^2=i,e^2=i,f^2=i,g^2=i,h^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,e^b=ei,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,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, 330389);