#426462

order	512
nilpotency class	3
isoclinism	isoclinic class of #418498
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2673
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15353
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

756 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 333658);