#147724

order	512
nilpotency class	4
isoclinism	isoclinic class of #140671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1022 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 1022 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54920);