#152858

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

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	C₂³	#508
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405 ◁ #2108 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

176 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 256

A minimal faithful permutation representation — 3 generators on 68 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=egi,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,d^b=di

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, 60054);