#482190

order	512
nilpotency class	4
isoclinism	isoclinic class of #481619
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

856 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)(3,10,12,15)(4,13)(5,8)(6,9)(7,11,16,14)(17,18,22,28,23,27,35,42)(19,25,30,39,20,24,32,37)(21,26,33,41,34,40,45,48)(29,38,43,47,31,36,44,46);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,23,20)(18,24,27,25)(21,31,34,29)(22,30,35,32)(26,38,40,36)(28,39,42,37)(33,44,45,43)(41,46,48,47);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,23,19)(18,25,27,24)(21,29,34,31)(22,30,35,32)(26,36,40,38)(28,37,42,39)(33,44,45,43)(41,47,48,46);(17,21,23,34)(18,26,27,40)(19,29,20,31)(22,33,35,45)(24,36,25,38)(28,41,42,48)(30,43,32,44)(37,46,39,47)

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,d^c=di,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, 389386);