#497346

order	512
nilpotency class	4
isoclinism	isoclinic class of #497333
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	D₄×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#61087	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#577 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1119 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,d^b=de,e^b=ei,f^c=fi,g^c=ghi,e^d=ei,g^f=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, 404542);