#130040

order	512
nilpotency class	4
isoclinism	isoclinic class of #130040
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8342
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#41	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

551 conjugacy classes of subgroups; only the 120 normal subgroups are shown.

normal subgroups only

This group has 551 conjugacy classes of subgroups — too many to draw. Only its 120 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=hi,e^2=i,f^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=dh,f^b=fgh,g^b=gi,e^c=ei,f^c=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, 37236);