#423855

order	512
nilpotency class	3
isoclinism	isoclinic class of #423854
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15316
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3068	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1287 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331051);