#132129

order	512
nilpotency class	4
isoclinism	isoclinic class of #132007
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#392	D₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2298 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

465 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 465 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,f^c=fi,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, 39325);