#421697

order	512
nilpotency class	3
isoclinism	isoclinic class of #418470
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2676
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15296
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

821 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,c^2=e,d^2=e,e^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^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, 328893);