#134864

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	#8620
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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=g,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^b=fi,f^c=fh,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287633, #72343286, #72343287
maximal normal subgroup of	#72287633, #72343286, #72343287

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42060);