#478497

order	512
nilpotency class	3
isoclinism	isoclinic class of #476443
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2829
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17151
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

777 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=efg,b^2=ef,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,e^c=ei,h^c=hi,f^d=fi,g^d=gi

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, 385693);