#152857

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=eg,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, 60053);