#495540

order	512
nilpotency class	4
isoclinism	isoclinic class of #33691
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

799 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=i,d^2=fh,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,e^b=eh,f^b=fi,g^b=gh,h^b=hi,e^c=ei,g^c=gi,e^d=ei,g^d=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, 402736);