#418469

order	512
nilpotency class	3
isoclinism	isoclinic class of #418466
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1651 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:

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

Polycyclic presentation:

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

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, 325665);