#151987

order	512
nilpotency class	3
isoclinism	isoclinic class of AGL(1,ℤ/8)
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₄	C₂×C₄×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	D₄×C₁₆
prime radical	C₂³	#504
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

214 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 146 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=de,d^2=i,e^2=g,g^2=h,h^2=i,b^a=bd,c^a=ci,d^b=di,e^b=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286543, #72287226, #72287227
maximal normal subgroup of	#72286543, #72287226, #72287227

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59183);