#139304

order	512
nilpotency class	4
isoclinism	isoclinic class of #139220
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

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	C₂×#548	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#548 ◁ G
LECS	G ▷ C₂×C₄×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

558 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 72 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=gh,d^2=gh,e^2=gh,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gi,h^a=hi,d^b=dh,f^b=fh,g^b=gi,h^b=hi,e^c=egh,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, 46500);