#470187

order	512
nilpotency class	3
isoclinism	isoclinic class of #455198
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2620
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16678
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

614 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377383);