#194693

order	512
nilpotency class	3
isoclinism	isoclinic class of #18346
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

1129 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9,10,16)(4,8)(5,7)(6,11,14,12)(13,15)(17,18)(19,23)(20,30,32,24)(21,33)(22,26)(25,34)(27,35,40,43)(28,41)(29,37)(31,38,45,44)(36,42)(39,47,48,46);(1,3,4,12,5,10,13,11)(2,6,8,16,7,14,15,9)(17,19,22,29)(18,23,26,37)(20,27,32,40)(21,28,34,42)(24,35,30,43)(25,36,33,41)(31,39,45,48)(38,46,44,47);(17,20,22,32)(18,24,26,30)(19,27,29,40)(21,31,34,45)(23,35,37,43)(25,38,33,44)(28,39,42,48)(36,46,41,47);(2,7)(3,10)(8,15)(11,12)(17,21,22,34)(18,25,26,33)(19,28,29,42)(20,31,32,45)(23,36,37,41)(24,38,30,44)(27,39,40,48)(35,46,43,47)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101889);