#500844

order	512
nilpotency class	4
isoclinism	isoclinic class of #500843
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	#525	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #2929 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

939 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 939 conjugacy classes of subgroups — too many to draw. Only its 166 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408040);