#296379

order	512
nilpotency class	3
isoclinism	isoclinic class of #269019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2617	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

910 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ei,c^2=fh,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 203575);