#152821

order	512
nilpotency class	5
isoclinism	isoclinic class of #13691
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2206
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂³	#509
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

212 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 74 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=d,d^2=i,e^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eg,g^a=gh,h^a=hi,e^c=egi,g^c=gh,h^c=hi

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, 60017);