#194683

order	512
nilpotency class	3
isoclinism	isoclinic class of #18316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1175 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egi,d^2=i,e^2=h,b^a=be,c^a=cf,d^a=dhi,f^a=fi,e^b=eh,f^b=fi,g^b=gh,e^c=ei,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, 101879);