#118391

order	512
nilpotency class	3
isoclinism	isoclinic class of #118389
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

790 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

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

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, 25587);