#132078

order	512
nilpotency class	4
isoclinism	isoclinic class of #132074
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#8459
commutator	#117	C₂²×C₄
Frattini, cosocle	#405	C₂³
prime radical	#3259	C₄
LCS	G ▷ #117 ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

496 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=hi,d^2=h,e^2=i,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,d^b=dh,e^b=ehi,f^b=fghi,g^b=gi,d^c=dh,e^d=ei,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39274);