#423203

order	512
nilpotency class	3
isoclinism	isoclinic class of #417732
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

1287 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)(7,19)(8,25)(9,15)(10,17)(11,16)(12,18)(13,22)(14,29)(20,49)(21,48)(23,38)(24,37)(26,47)(27,57)(28,42)(30,41)(31,62)(32,45)(33,44)(34,46)(35,52)(36,53)(39,51)(40,60)(43,54)(50,64)(55,61)(56,65)(58,68)(59,66)(63,67);(5,7,12,24)(6,13,18,38)(8,26,30,51)(9,31,34,53)(10,23,11,22)(14,39,42,47)(15,43,46,48)(16,37,17,19)(20,28,52,29)(21,33,54,32)(25,49,41,35)(27,58,61,50)(36,45,62,44)(40,64,66,68)(55,65,57,67)(56,60,63,59);(1,2,3,4)(5,8)(6,14)(7,20)(9,27)(10,29)(11,28)(12,30)(13,35)(15,40)(16,25)(17,41)(18,42)(19,47)(21,50)(22,26)(23,51)(24,52)(31,63)(32,60)(33,59)(34,61)(36,64)(37,39)(38,49)(43,65)(44,57)(45,55)(46,66)(48,67)(53,56)(54,58)(62,68);(5,9,12,34)(6,15,18,46)(7,21,24,54)(8,27,30,61)(10,33,11,32)(13,36,38,62)(14,40,42,66)(16,45,17,44)(19,48,37,43)(20,50,52,58)(22,53,23,31)(25,55,41,57)(26,56,51,63)(28,60,29,59)(35,64,49,68)(39,65,47,67)

Polycyclic presentation:

2^9|b^2=i,c^2=ef,d^2=i,e^2=i,f^2=i,g^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,g^b=gi,e^c=ei,f^c=fi,g^c=gi,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, 330399);