#494848

order	512
nilpotency class	3
isoclinism	isoclinic class of #23472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#124
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1924
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

735 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,c^b=ch,e^b=ei,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402044);