#482823

order	512
nilpotency class	4
isoclinism	isoclinic class of #482722
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	C₂²×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#557 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

767 conjugacy classes of subgroups; only the 198 normal subgroups are shown.

normal subgroups only

This group has 767 conjugacy classes of subgroups — too many to draw. Only its 198 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=hi,e^2=h,f^2=i,b^a=be,e^a=eg,g^a=gi,e^b=eh,f^b=fgh,d^c=dhi,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, 390019);