#493491

order	512
nilpotency class	4
isoclinism	isoclinic class of #493483
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#20585
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	#62887	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ #3487 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400687);