#479823

order	512
nilpotency class	3
isoclinism	isoclinic class of #477835
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2793
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17268
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

500 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:

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

Polycyclic presentation:

2^9|a^2=efg,b^2=efh,c^2=ef,d^2=e,e^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=def,e^b=ei,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,g^d=gi

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, 387019);