#421487

order	512
nilpotency class	3
isoclinism	isoclinic class of #418473
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	#15292
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

864 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,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,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, 328683);