#411715

order	512
nilpotency class	3
isoclinism	isoclinic class of #400051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1794 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)(5,6,12,18)(7,19,23,44)(8,14,28,41)(9,29,32,42)(10,17,11,16)(13,33,36,22)(15,30,43,31)(20,37,49,45)(21,50,52,35)(24,34,48,60)(25,54,57,62)(26,40,27,39)(38,55,63,56)(46,58,64,51)(47,59,67,68)(53,65,66,61);(2,4)(5,7)(6,13)(8,24)(9,21)(10,33)(11,22)(12,23)(14,37)(15,35)(16,44)(17,19)(18,36)(20,27)(25,53)(26,49)(28,48)(29,46)(30,58)(31,51)(32,52)(34,40)(38,61)(39,60)(41,45)(42,64)(43,50)(47,56)(54,59)(55,67)(57,66)(62,68)(63,65);(5,8)(6,14)(7,20)(9,25)(10,26)(11,27)(12,28)(13,34)(15,38)(16,39)(17,40)(18,41)(19,45)(21,47)(22,48)(23,49)(24,33)(29,54)(30,55)(31,56)(32,57)(35,59)(36,60)(37,44)(42,62)(43,63)(46,65)(50,68)(51,66)(52,67)(53,58)(61,64);(5,9,10,30,12,32,11,31)(6,15,16,42,18,43,17,29)(7,21,33,58,23,52,22,51)(8,25,26,55,28,57,27,56)(13,35,44,64,36,50,19,46)(14,38,39,62,41,63,40,54)(20,47,24,53,49,67,48,66)(34,59,37,61,60,68,45,65)

Polycyclic presentation:

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

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, 318911);