#419008

order	512
nilpotency class	3
isoclinism	isoclinic class of #418976
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3020	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2189 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)(3,4)(5,6,10,17,12,18,11,16)(7,19,22,13,23,37,34,38)(8,24,28,42,29,41,27,14)(9,15,32,44,33,46,31,45)(20,48,25,55,51,35,50,39)(21,47,52,36,53,43,30,61)(26,54,58,65,60,64,59,40)(49,67,56,66,57,62,68,63);(5,7,10,22,12,23,11,34)(6,13,16,19,18,38,17,37)(8,25,27,20,29,50,28,51)(9,30,31,53,33,52,32,21)(14,39,24,48,42,55,41,35)(15,43,44,61,46,47,45,36)(26,57,58,68,60,49,59,56)(40,62,64,66,65,67,54,63);(5,8,12,29)(6,14,18,42)(7,20,23,51)(9,26,33,60)(10,27,11,28)(13,35,38,48)(15,40,46,65)(16,24,17,41)(19,39,37,55)(21,49,53,57)(22,50,34,25)(30,56,52,68)(31,58,32,59)(36,62,61,67)(43,66,47,63)(44,64,45,54);(2,4)(5,9)(6,15)(7,21)(8,26)(10,32)(11,31)(12,33)(13,36)(14,40)(16,45)(17,44)(18,46)(19,47)(20,49)(22,52)(23,53)(24,54)(25,56)(27,59)(28,58)(29,60)(30,34)(35,62)(37,43)(38,61)(39,63)(41,64)(42,65)(48,67)(50,68)(51,57)(55,66)

Polycyclic presentation:

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

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, 326204);