#493495

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	#62833	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #3499 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

2178 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)(3,13,16,30)(4,10)(5,18,20,8)(6,11)(7,12)(9,15,27,21)(14,29)(17,28)(19,26,32,25)(22,24)(23,31);(2,8)(9,24)(10,13)(11,25)(12,18)(26,31)(27,28)(29,30);(1,3,7,16)(2,9,12,27)(4,5,14,20)(6,21,23,15)(8,24,18,28)(10,25,29,26)(11,30,31,13)(17,32,22,19);(1,4,7,14)(2,10,12,29)(3,5,16,20)(6,22,23,17)(8,13,18,30)(9,26,27,25)(11,24,31,28)(15,32,21,19)

Polycyclic presentation:

2^9|a^2=g,c^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, 400691);