#509612

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13307	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

636 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)(7,18)(8,24)(9,31)(10,36)(11,17)(12,20)(13,21)(14,44)(15,34)(16,33)(19,37)(22,43)(23,42)(25,41)(26,51)(27,48)(28,47)(29,59)(30,60)(32,38)(35,46)(39,52)(40,55)(45,61)(49,64)(50,63)(53,58)(54,66)(56,57)(62,68)(65,67);(1,2)(3,4)(5,7)(6,12)(8,25)(9,32)(10,21)(11,23)(13,15)(14,45)(16,38)(17,40)(18,31)(19,27)(20,34)(22,33)(24,57)(26,55)(28,61)(29,65)(30,62)(35,52)(36,39)(37,47)(41,48)(42,46)(43,51)(44,53)(49,68)(50,67)(54,64)(56,63)(58,60)(59,66);(2,4)(5,7)(6,13)(8,19)(9,20)(10,22)(11,23)(12,15)(14,41)(16,42)(17,43)(18,36)(21,33)(24,58)(25,27)(26,52)(28,56)(29,53)(30,54)(31,39)(32,34)(35,55)(37,47)(38,46)(40,51)(44,65)(45,48)(49,67)(50,68)(57,60)(59,66)(61,63)(62,64);(5,8,26,44,10,29,9,27,11,30,34,63,33,28,35,64)(6,14,46,24,16,49,15,47,17,50,31,60,36,48,51,59)(7,19,52,65,22,53,20,25,23,54,32,61,21,56,55,62)(12,37,43,68,39,57,18,45,40,66,13,41,38,58,42,67)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416808);