#152836

order	512
nilpotency class	5
isoclinism	isoclinic class of #13700
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2215
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

218 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 78 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)(6,13)(7,20)(8,14)(9,27)(10,25)(11,26)(15,42)(16,46)(17,48)(18,49)(19,37)(21,52)(22,53)(23,38)(24,29)(28,41)(30,39)(31,40)(32,43)(33,35)(34,36)(44,66)(45,61)(47,59)(50,63)(51,55)(54,64)(57,58)(60,62)(65,67);(1,3,4,2)(5,6,14,35,10,17,40,49,26,34,9,16,39,60,28,37,12,19,41,62,30,46,27,36,11,18,31,48,25,33,8,13)(7,15,38,57,22,44,63,67,55,61,29,47,64,68,54,59,24,45,51,65,50,66,53,58,23,42,20,43,52,56,21,32);(5,7,8,21,25,52,31,20,11,23,27,53,30,50,41,51,12,24,28,54,39,64,9,29,26,55,40,63,10,22,14,38)(6,15,13,32,33,56,48,43,18,42,36,58,46,66,62,65,19,45,37,59,60,68,16,47,34,61,49,67,17,44,35,57)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=d,d^2=gh,e^2=gi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,g^a=gh,h^a=hi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60032);