#483028

order	512
nilpotency class	4
isoclinism	isoclinic class of #483025
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17675
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#97	C₂⁴
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ #97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

644 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 644 conjugacy classes of subgroups — too many to draw. Only its 188 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390224);