#481883

order	512
nilpotency class	4
isoclinism	isoclinic class of #481837
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

841 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)(3,9,11,16)(4,8)(5,7)(6,10,14,12)(13,15)(17,18,22,28,23,27,30,39)(19,24,32,40,31,41,21,26)(20,33,35,48,36,25,42,44)(29,46,34,45,47,38,37,43);(1,3)(2,6)(4,12)(5,11)(7,14)(8,16)(9,15)(10,13)(17,19)(18,24,27,41)(20,29)(21,30)(22,32)(23,31)(25,38,33,46)(26,39,40,28)(34,35)(36,47)(37,42)(43,44,45,48);(2,7)(6,14)(8,15)(9,16)(17,20,23,36)(18,25,27,33)(19,29,31,47)(21,37,32,34)(22,35,30,42)(24,38,41,46)(26,45,40,43)(28,44,39,48);(17,21,23,32)(18,26,27,40)(19,30,31,22)(20,34,36,37)(24,39,41,28)(25,43,33,45)(29,35,47,42)(38,44,46,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,e^b=eh,f^b=fghi,d^c=di,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, 389079);