#480746

order	512
nilpotency class	4
isoclinism	isoclinic class of #480674
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1112 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,e^2=i,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=ch,f^b=fgi,f^e=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387942);