#397203

order	512
nilpotency class	3
isoclinism	isoclinic class of #394823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
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

1407 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=ei,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,h^a=hi,c^b=cg,g^b=gi,g^c=gi,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 304399);