#150721

order	512
nilpotency class	5
isoclinism	isoclinic class of #150719
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11829
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	#423	C₂³
prime radical	#575	C₂×C₄
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2960 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

224 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 52 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)(3,13,16,34,18,40,17,49)(4,19)(5,10,26,36)(6,28)(7,11)(8,15,30,47,31,48,38,24)(9,22)(12,20)(14,41,43,61,45,64,44,39)(21,51,25,58)(23,52)(27,60,54,37)(29,57,62,46,63,55,50,42)(32,59,35,56)(33,53);(1,3)(2,8)(4,14)(5,24)(6,16)(7,18)(9,29)(10,34)(11,31)(12,38)(13,37)(15,27)(17,20)(19,50)(21,55)(22,43)(23,45)(25,57)(26,47)(28,30)(32,64)(33,63)(35,41)(36,49)(39,58)(40,60)(42,56)(44,53)(46,59)(48,54)(51,61)(52,62);(1,4,20,53,7,23,6,22)(2,9,28,52,11,33,12,19)(3,14,17,44,18,45,16,43)(5,25,54,56,26,21,27,59)(8,29,30,62,31,63,38,50)(10,35,60,51,36,32,37,58)(13,39,34,41,40,61,49,64)(15,46,24,57,48,42,47,55)

Polycyclic presentation:

2^9|a^2=e,c^2=gh,d^2=gi,e^2=hi,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gi,d^b=dg,e^b=efg,g^b=gi,h^b=hi,d^c=di,e^d=eh,f^d=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, 57917);