#479252

order	512
nilpotency class	3
isoclinism	isoclinic class of #477835
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2793
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17194
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

643 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:

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

Polycyclic presentation:

2^9|a^2=efg,c^2=e,d^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,d^c=de,g^c=gi,h^c=hi,e^d=ei,g^d=gi,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, 386448);