#493501

order	512
nilpotency class	4
isoclinism	isoclinic class of #493500
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#20585
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#62876	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #3500 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2269 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)(3,12,15,7)(4,9)(5,10)(6,11)(8,14,23,18)(13,24,30,21)(16,27)(17,26)(19,28)(20,29)(22,32,25,31);(2,7)(4,16)(8,10)(9,24)(11,12)(13,30)(17,19)(21,27)(22,26)(23,29)(25,28)(31,32);(1,3)(2,8)(4,13)(5,18)(6,15)(7,10)(9,25)(11,23)(12,29)(14,20)(16,30)(17,32)(19,31)(21,26)(22,27)(24,28);(1,4,6,16)(2,9,11,27)(3,13,15,30)(5,19,20,17)(7,21,12,24)(8,22,23,25)(10,28,29,26)(14,32,18,31)

Polycyclic presentation:

2^9|a^2=g,d^2=i,f^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,g^b=gh,d^c=dh,f^c=fi,g^c=gi,f^d=fi,g^e=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400697);