#423197

order	512
nilpotency class	3
isoclinism	isoclinic class of #417738
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
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

1292 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)(5,6,11,16)(7,17,21,42)(8,22,25,43)(9,14,28,41)(10,29,33,15)(12,30,36,20)(13,31,38,24)(18,34,49,63)(19,46,50,39)(23,54,57,53)(26,59,52,35)(27,44,32,40)(37,58,65,56)(45,51,62,48)(47,67,68,61)(55,64,60,66);(5,7,11,21)(6,12,16,36)(8,18,25,49)(9,26,28,52)(10,30,33,20)(13,34,38,63)(14,39,41,46)(15,42,29,17)(19,32,50,27)(22,45,43,62)(23,55,57,60)(24,51,31,48)(35,44,59,40)(37,64,65,66)(47,58,68,56)(53,67,54,61);(1,2,3,4)(5,8,11,25)(6,13,16,38)(7,18,21,49)(9,23,28,57)(10,31,33,24)(12,34,36,63)(14,37,41,65)(15,43,29,22)(17,45,42,62)(19,47,50,68)(20,51,30,48)(26,60,52,55)(27,58,32,56)(35,61,59,67)(39,66,46,64)(40,53,44,54);(5,9)(6,14)(7,19)(8,23)(10,32)(11,28)(12,35)(13,37)(15,44)(16,41)(17,46)(18,47)(20,52)(21,50)(22,53)(24,58)(25,57)(26,30)(27,33)(29,40)(31,56)(34,61)(36,59)(38,65)(39,42)(43,54)(45,64)(48,60)(49,68)(51,55)(62,66)(63,67)

Polycyclic presentation:

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

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, 330393);