#134868

order	512
nilpotency class	4
isoclinism	isoclinic class of #134748
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1812
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	#405	C₂³
socle	C₂	#8622
prime radical	#525	Q₈
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #2151 ◁ G
LECS	G ▷ #405 ▷ C₂×C₄ ▷ C₂ ▷ C₁

242 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dg,d^2=gi,f^2=gh,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci,f^c=fh,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287633, #72319344
maximal normal subgroup of	#72287633, #72319344

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42064);