#121097

order	512
nilpotency class	3
isoclinism	isoclinic class of #121095
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

372 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=hi,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=di,e^b=ei,g^b=gi,e^c=eh,f^c=fhi

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, 28293);