#472005

order	512
nilpotency class	3
isoclinism	isoclinic class of #465214
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

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

1298 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)(7,17,21,45)(8,22)(9,27,30,14)(10,15)(11,16)(12,20,35,53)(13,26)(18,43,47,36)(19,48)(23,33,52,63)(24,54,55,37)(25,41)(28,57,60,39)(29,58,61,40)(31,38)(32,42)(34,50)(44,51)(46,66)(49,62)(56,68)(59,64)(65,67);(1,2,3,4)(5,7,11,21)(6,12)(8,23)(9,19,30,51)(10,20,32,53)(13,36,22,43)(14,34)(15,17)(16,35)(18,31)(24,49)(25,47)(26,52)(27,56)(28,59)(29,50,61,68)(33,38,63,41)(37,64,54,66)(39,65,57,62)(40,44)(42,45)(46,60)(48,58)(55,67);(2,4)(5,8)(6,13)(7,18)(9,28)(10,31)(11,26)(12,33)(14,39)(15,41)(16,22)(17,43)(19,49)(20,52)(21,47)(23,53)(24,29)(25,32)(27,57)(30,60)(34,64)(35,63)(36,45)(37,40)(38,42)(44,62)(46,50)(48,65)(51,67)(54,58)(55,61)(56,66)(59,68);(5,9,10,29,11,30,32,61)(6,14,15,40,16,27,42,58)(7,19,20,50,21,51,53,68)(8,24,25,28,26,55,31,60)(12,34,17,44,35,56,45,48)(13,37,38,39,22,54,41,57)(18,46,23,49,47,59,52,67)(33,62,36,64,63,65,43,66)

Polycyclic presentation:

2^9|a^2=fi,b^2=egh,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,h^c=hi,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, 379201);