#198749

order	512
nilpotency class	3
isoclinism	isoclinic class of #198516
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1368 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,d^2=hi,e^2=hi,f^2=hi,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=di,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fhi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 105945);