#483236

order	512
nilpotency class	4
isoclinism	isoclinic class of #483235
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#1551
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

683 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,e^2=h,f^2=gh,g^2=i,b^a=be,c^a=cg,e^a=eg,g^a=gi,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,d^c=di,e^c=ei

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, 390432);