#479819

order	512
nilpotency class	3
isoclinism	isoclinic class of #478580
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2789
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17264
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

512 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:

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

Polycyclic presentation:

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

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, 387015);