#151017

order	512
nilpotency class	5
isoclinism	isoclinic class of #151015
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11982
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₄×C₁₆	C₂³
prime radical	#2218	C₄
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₄×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

189 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 189 conjugacy classes of subgroups — too many to draw. Only its 62 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=fh,d^2=gh,e^2=fghi,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=cf,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,g^b=gh,h^b=hi,d^c=di

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286285, #72319310, #72319344, …
maximal normal subgroup of	#72286285, #72319310, #72319344, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58213);