#150845

order	512
nilpotency class	5
isoclinism	isoclinic class of #150842
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11878
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	#408	C₂³
prime radical	#532	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

253 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 253 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=ghi,d^2=g,e^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cfh,d^b=dg,e^b=efgi,g^b=gi,d^c=dh,e^c=ei,f^c=fi,e^d=eh,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58041);