#493497

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	#62836	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #3500 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2229 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)(3,15)(4,19,21,31)(5,11)(6,23,25,12)(7,13)(8,14)(9,18,28,24)(10,16,29,26)(17,27)(20,32)(22,30);(1,3,8,7)(2,9,14,28)(4,16,21,26)(5,17,22,20)(6,18,25,24)(10,27,29,32)(11,19,30,31)(12,15,23,13);(1,4)(2,10)(3,16)(5,18)(6,20)(7,26)(8,21)(9,27)(11,12)(13,31)(14,29)(15,19)(17,25)(22,24)(23,30)(28,32);(1,5,8,22)(2,11,14,30)(3,17,7,20)(4,18,21,24)(6,16,25,26)(9,19,28,31)(10,23,29,12)(13,32,15,27)

Polycyclic presentation:

2^9|a^2=g,b^2=i,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, 400693);