#494833

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₂×#121
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1923
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

767 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402029);