#511964

order	512
nilpotency class	4
isoclinism	isoclinic class of #511963
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

928 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 186 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 — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,h^a=hi,d^b=di,f^c=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419160);