#355151

order	512
nilpotency class	3
isoclinism	isoclinic class of #21782
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2308
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13776
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1239 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)(3,14,17,8)(4,18,20,10)(5,12)(6,13)(7,11)(9,19,27,21)(15,24)(16,28,22,25)(23,30)(26,32)(29,31);(1,3,6,16,7,17,5,22)(2,8,13,25,11,14,12,28)(4,15,21,30,20,31,19,32)(9,26,10,24,27,23,18,29);(1,4)(2,9,11,27)(3,15)(5,19)(6,21)(7,20)(8,23,14,26)(10,12,18,13)(16,30)(17,31)(22,32)(24,25,29,28);(1,4)(2,10)(3,15)(5,19)(6,21)(7,20)(8,24)(9,12)(11,18)(13,27)(14,29)(16,30)(17,31)(22,32)(23,25)(26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262347);