#140728

order	512
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1970
prime radical	C₂³	C₄×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1450 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

328 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 328 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 72
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2,4,7,5,3,6,8)(9,10)(11,23,28,47)(12,30)(13,20)(14,37)(15,39)(16,22)(17,26,42,35)(18,32)(19,24)(21,38)(25,40,55,66)(27,43,51,65)(29,41,57,50)(31,59)(33,44)(34,61)(36,48)(45,63)(46,52)(49,69,68,54)(53,70,62,58)(56,72,71,67)(60,64);(1,3,4,8,5,2,6,7)(9,11,24,51,38,57,13,35,16,28,14,27,15,29,36,26)(10,17,20,41,39,65,19,47,22,42,48,50,21,43,37,23)(12,25,52,69,64,71,31,62,34,55,32,54,33,56,63,53)(18,40,30,58,59,72,44,68,46,66,61,70,45,67,60,49);(9,12,13,31,15,33,24,52,16,34,36,63,38,64,14,32)(10,18,37,60,21,45,48,61,22,46,19,44,39,59,20,30)(11,25,35,62,29,56,51,69,28,55,26,53,57,71,27,54)(17,40,23,49,43,67,50,70,42,66,47,68,65,72,41,58)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=eg,d^2=g,e^2=gi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,d^b=di,f^b=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47924);