#395415

order	512
nilpotency class	3
isoclinism	isoclinic class of #394055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#578	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3367 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)(5,6,10,18,12,16,11,17)(7,19,22,37,24,13,23,38)(8,25,33,42,28,41,29,14)(9,30,34,15,31,43,32,44)(20,39,52,35,48,61,26,45)(21,49,53,46,50,62,51,36)(27,56,57,54,59,40,58,64)(47,60,55,63,67,65,66,68);(2,4)(5,7,12,24)(6,13,16,19)(8,26,28,52)(9,21,31,50)(10,23,11,22)(14,39,42,61)(15,36,44,46)(17,38,18,37)(20,33,48,29)(25,45,41,35)(27,55,59,66)(30,49,43,62)(32,53,34,51)(40,63,56,68)(47,58,67,57)(54,65,64,60);(5,8,12,28)(6,14,16,42)(7,20,24,48)(9,27,31,59)(10,33,11,29)(13,35,19,45)(15,40,44,56)(17,41,18,25)(21,47,50,67)(22,52,23,26)(30,54,43,64)(32,58,34,57)(36,60,46,65)(37,39,38,61)(49,63,62,68)(51,66,53,55);(5,9)(6,15)(7,21)(8,27)(10,32)(11,34)(12,31)(13,36)(14,40)(16,44)(17,43)(18,30)(19,46)(20,47)(22,51)(23,53)(24,50)(25,54)(26,55)(28,59)(29,57)(33,58)(35,60)(37,49)(38,62)(39,63)(41,64)(42,56)(45,65)(48,67)(52,66)(61,68)

Polycyclic presentation:

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

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, 302611);