#481443

order	512
nilpotency class	4
isoclinism	isoclinic class of #481438
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#327
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	#60461	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×#41 ◁ C₂×#568 ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

1172 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

normal subgroups only

This group has 1172 conjugacy classes of subgroups — too many to draw. Only its 272 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) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,5,11,7,12,6,20)(3,9,15,26,13,24,16,30)(4,8,17,10,19,23,18,22)(14,21,28,25,27,31,29,32);(2,8)(4,17)(5,6)(9,21)(10,11)(12,23)(14,28)(15,16)(18,19)(20,22)(24,31)(25,26)(27,29)(30,32);(1,3)(2,9)(4,14)(5,15)(6,16)(7,13)(8,21)(10,25)(11,26)(12,24)(17,28)(18,29)(19,27)(20,30)(22,32)(23,31);(3,13)(9,24)(14,27)(15,16)(21,31)(25,32)(26,30)(28,29)

Polycyclic presentation:

2^9|a^2=f,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,e^a=eg,h^a=hi,e^b=eh,f^b=fghi,g^b=gi,h^b=hi,d^c=di,f^e=fi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388639);