#411417

order	512
nilpotency class	3
isoclinism	isoclinic class of #405373
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2853
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15116
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

621 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:

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

Polycyclic presentation:

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

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, 318613);