#492958

order	512
nilpotency class	3
isoclinism	isoclinic class of #10792
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	C₄×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂	C₂×#1749
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1193 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)(3,13,15,8)(4,9)(5,11)(6,12)(7,10)(14,37,41,26)(16,38,43,27)(17,39,44,28)(18,47,48,29)(19,30)(20,31)(21,32)(22,35)(23,36)(24,33)(25,34)(40,54)(42,56,45,55)(46,63,64,57)(49,62,52,58)(50,59)(51,60)(53,61);(1,3,5,16,6,17,23,46,7,15,24,43,25,44,53,64)(2,8,11,27,12,28,36,57,10,13,33,38,34,39,61,63)(4,18,20,49,21,41,51,45,19,48,22,52,50,14,40,42)(9,29,31,58,32,37,60,56,30,47,35,62,59,26,54,55);(1,4)(2,9)(3,14,15,41)(5,22)(6,21)(7,19)(8,26,13,37)(10,30)(11,35)(12,32)(16,45,43,42)(17,18,44,48)(20,24)(23,40)(25,50)(27,56,38,55)(28,29,39,47)(31,33)(34,59)(36,54)(46,52,64,49)(51,53)(57,62,63,58)(60,61);(4,19)(9,30)(14,41)(18,48)(20,22)(21,50)(26,37)(29,47)(31,35)(32,59)(40,51)(42,45)(49,52)(54,60)(55,56)(58,62)

Polycyclic presentation:

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

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, 400154);