#152423

order	512
nilpotency class	5
isoclinism	isoclinic class of #13669
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2184
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
prime radical	C₂×#41	D₈
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#131 ◁ C₂×#504 ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

263 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 263 conjugacy classes of subgroups — too many to draw. Only its 94 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)(5,6)(7,20)(8,25)(9,18)(10,34)(11,16)(12,19)(13,21)(14,22)(15,29)(17,27)(23,54)(24,47)(26,49)(28,37)(30,56)(31,57)(32,42)(33,50)(35,43)(36,48)(38,59)(39,58)(40,53)(41,55)(44,61)(45,62)(46,63)(51,52)(60,68)(64,66)(65,67);(1,3,4,2)(5,7)(6,13)(8,26)(9,22)(10,35)(11,32)(12,24)(14,37)(15,43)(16,38)(17,49)(18,51)(19,40)(20,25)(21,29)(23,27)(28,58)(30,64)(31,60)(33,55)(34,39)(36,65)(41,45)(42,56)(44,54)(46,67)(47,57)(48,66)(50,68)(52,62)(53,63)(59,61);(5,7)(6,14)(8,21)(9,32)(10,23)(11,22)(12,24)(13,37)(15,39)(16,47)(17,41)(18,20)(19,42)(25,51)(26,29)(27,35)(28,59)(30,52)(31,53)(33,65)(34,43)(36,55)(38,57)(40,56)(44,68)(45,49)(46,66)(48,67)(50,54)(58,61)(60,63)(62,64)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59619);