#391012

order	512
nilpotency class	3
isoclinism	isoclinic class of #32696
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2749
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14597
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

624 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298208);