#482024

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	#2619	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#519 ◁ 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) = 64

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

Polycyclic presentation:

2^9|a^2=f,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,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, 389220);