#482184

order	512
nilpotency class	4
isoclinism	isoclinic class of #481601
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	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

893 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)(3,10,12,15)(4,13)(5,8)(6,9)(7,11,16,14)(17,18,22,27,23,26,36,42)(19,28,30,43,31,24,37,40)(20,32,33,44,21,25,35,41)(29,38,45,47,34,39,46,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,23,31)(18,24,26,28)(20,29,21,34)(22,30,36,37)(25,38,32,39)(27,43,42,40)(33,45,35,46)(41,48,44,47);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,23,21)(18,25,26,32)(19,29,31,34)(22,35,36,33)(24,38,28,39)(27,44,42,41)(30,46,37,45)(40,48,43,47);(17,21,23,20)(18,25,26,32)(19,29,31,34)(22,35,36,33)(24,39,28,38)(27,41,42,44)(30,45,37,46)(40,48,43,47)

Polycyclic presentation:

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

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, 389380);