#481984

order	512
nilpotency class	4
isoclinism	isoclinic class of #11799
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
socle	C₂²	C₂²×#99
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

842 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)(3,11,13,16)(4,9)(5,10)(6,8)(7,12,15,14)(17,18,22,28,23,27,36,43)(19,29,31,44,32,24,37,39)(20,25,34,41,21,26,35,42)(30,40,45,48,33,38,46,47);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,23,32)(18,24,27,29)(20,33,21,30)(22,31,36,37)(25,40,26,38)(28,44,43,39)(34,46,35,45)(41,47,42,48);(2,8)(7,15)(9,10)(11,16)(17,20,23,21)(18,25,27,26)(19,30,32,33)(22,34,36,35)(24,38,29,40)(28,41,43,42)(31,45,37,46)(39,47,44,48);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(17,21,23,20)(18,26,27,25)(19,30,32,33)(22,35,36,34)(24,38,29,40)(28,42,43,41)(31,45,37,46)(39,47,44,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,f^e=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389180);