#395189

order	512
nilpotency class	3
isoclinism	isoclinic class of #395187
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
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

1692 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)(8,25,29,39)(9,30,33,41)(10,17,11,16)(13,23,37,22)(14,27,40,28)(15,31,42,32)(20,34,48,59)(21,35,51,60)(26,38,55,62)(43,46,52,47)(44,49,57,50)(45,63,65,67)(53,56,61,54)(58,66,68,64);(1,2)(3,4)(5,7,11,23,12,24,10,22)(6,13,17,36,18,37,16,19)(8,20,28,47,29,48,27,46)(9,21,32,50,33,51,31,49)(14,34,25,52,40,59,39,43)(15,35,30,57,42,60,41,44)(26,45,56,66,55,65,54,64)(38,58,53,67,62,68,61,63);(2,4)(5,8)(6,14)(7,20)(9,26)(10,27)(11,28)(12,29)(13,34)(15,38)(16,39)(17,25)(18,40)(19,43)(21,45)(22,46)(23,47)(24,48)(30,53)(31,54)(32,56)(33,55)(35,58)(36,52)(37,59)(41,61)(42,62)(44,63)(49,64)(50,66)(51,65)(57,67)(60,68);(5,9,10,31,12,33,11,32)(6,15,16,41,18,42,17,30)(7,21,22,49,24,51,23,50)(8,26,27,54,29,55,28,56)(13,35,19,44,37,60,36,57)(14,38,39,61,40,62,25,53)(20,45,46,64,48,65,47,66)(34,58,43,63,59,68,52,67)

Polycyclic presentation:

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

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, 302385);