#493493

order	512
nilpotency class	4
isoclinism	isoclinic class of #493492
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	#62824	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #3500 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2248 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)(13,29,31,21)(16,25)(17,26)(19,27)(20,28)(22,32,24,30);(2,7)(8,10)(9,21)(11,12)(22,27)(23,28)(24,25)(26,29);(1,3)(2,8)(4,13)(5,18)(6,15)(7,10)(9,24)(11,23)(12,28)(14,20)(16,32)(17,31)(19,30)(21,25)(22,26)(27,29);(1,4,6,17)(2,9,11,26)(3,13,15,31)(5,19,20,16)(7,21,12,29)(8,22,23,24)(10,27,28,25)(14,32,18,30)

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,g^b=gh,d^c=dh,f^c=fi,g^c=gi,f^d=fi,g^e=gi

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, 400689);