#139261

order	512
nilpotency class	4
isoclinism	isoclinic class of #139220
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1898
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×C₄×D₄	C₂³
prime radical	#2408	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#552 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

527 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 527 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=i,d^2=ghi,e^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,g^a=gi,h^a=hi,d^b=dh,f^b=fhi,g^b=gi,h^b=hi,e^c=eghi,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 46457);