#481438

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

1192 conjugacy classes of subgroups; only the 272 normal subgroups are shown.

normal subgroups only

This group has 1192 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)(3,8)(4,16,19,7)(5,10)(6,12)(9,17,21,30)(11,18)(13,22)(14,28,26,20)(15,24)(23,29,31,32)(25,27);(2,7)(4,17)(5,18)(8,20)(9,11)(10,21)(12,16)(14,29)(15,27)(19,30)(22,28)(23,25)(24,31)(26,32);(1,3)(2,8)(4,14)(5,15)(6,13)(7,20)(9,23)(10,24)(11,25)(12,22)(16,28)(17,29)(18,27)(19,26)(21,31)(30,32);(3,13)(8,22)(14,26)(15,27)(20,28)(23,31)(24,25)(29,32)

Polycyclic presentation:

2^9|a^2=f,e^2=hi,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, 388634);