#153592

order	512
nilpotency class	6
isoclinism	isoclinic class of #13742
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₃₂
commutator	C₃₂	C₂²×C₄
Frattini, cosocle	C₂×C₃₂	C₂³
prime radical	C₂×D₃₂	C₂²
LCS	G ▷ C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂ ◁ G
LECS	G ▷ C₂×C₃₂ ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

120 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 120 conjugacy classes of subgroups — too many to draw. Only its 50 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)(7,12)(8,23)(9,27)(10,30)(11,13)(14,18)(15,19)(16,24)(17,28)(20,41)(21,42)(22,31)(25,58)(26,54)(29,60)(32,44)(33,43)(34,51)(35,46)(36,49)(37,57)(38,50)(39,53)(40,59)(45,61)(47,56)(48,55)(52,65)(62,67)(63,66)(64,68);(6,12)(7,18)(8,19)(9,24)(10,28)(13,31)(14,15)(16,41)(17,42)(20,50)(21,53)(22,46)(23,32)(25,59)(26,49)(27,33)(29,57)(30,34)(35,36)(37,65)(38,56)(39,55)(40,61)(43,44)(45,51)(47,48)(52,68)(54,62)(58,64)(60,63)(66,67);(1,3,4,2)(6,13)(12,31)(14,35)(15,36)(16,37)(17,30)(23,54)(27,60)(32,62)(33,63)(34,42)(38,64)(39,61)(40,55)(41,65)(56,58)

Polycyclic presentation:

2^9|a^2=e,c^2=e,d^2=fg,f^2=gh,g^2=hi,h^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fg,g^a=gh,h^a=hi,d^b=df,f^b=fg,g^b=gh,h^b=hi

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, 60788);