#150814

order	512
nilpotency class	5
isoclinism	isoclinic class of #150812
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11875
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	#408	C₂³
prime radical	#28177	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2932 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

298 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 58 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=gh,d^2=gi,e^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cf,d^b=dg,e^b=efg,g^b=gi,d^c=dhi,e^c=ei,f^c=fi,e^d=eh,f^e=fi

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58010);