#139738

order	512
nilpotency class	4
isoclinism	isoclinic class of #139736
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9495
commutator	C₂²×Q₈	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	#23917	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂³×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

642 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 66 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46934);