#447696

order	512
nilpotency class	3
isoclinism	isoclinic class of #427666
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2527
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15884
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

634 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)(3,8,14,24)(4,16,17,25)(5,9)(6,12)(7,11)(10,19,27,18)(13,28)(15,21,20,23)(22,32)(26,29)(30,31);(1,3,5,20,7,14,6,15)(2,8,12,21,11,24,9,23)(4,13,19,29,17,30,18,32)(10,26,25,28,27,22,16,31);(2,9,11,12)(8,21,24,23)(10,25,27,16)(22,28,26,31);(1,4,5,18,7,17,6,19)(2,10,12,16,11,27,9,25)(3,13,15,29,14,30,20,32)(8,22,23,31,24,26,21,28)

Polycyclic presentation:

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

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, 354892);