#150327

order	512
nilpotency class	5
isoclinism	isoclinic class of #150171
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#353
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	#407	C₂³
socle	C₂	#11726
prime radical	#23794	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2929 ◁ G
LECS	G ▷ #407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

339 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 339 conjugacy classes of subgroups — too many to draw. Only its 54 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 — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=gh,d^2=gi,f^2=i,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,h^b=hi,e^d=ehi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57523);