#482194

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×C₂²⋊C₄	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×D₄ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

952 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)(19,27)(20,31,32,24)(21,34,35,25)(22,26)(23,29,37,44)(28,41,42,39)(30,40)(33,38)(36,43)(45,48)(46,47);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,22,30)(18,23,26,37)(20,28,32,42)(21,29,35,44)(24,36,31,45)(25,27,34,40)(33,43,47,48)(38,39,46,41);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20)(18,24)(19,28)(21,33)(22,32)(23,36)(25,38)(26,31)(27,39)(29,43)(30,42)(34,46)(35,47)(37,45)(40,41)(44,48);(20,32)(24,31)(28,42)(33,47)(36,45)(38,46)(39,41)(43,48)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,e^2=h,b^a=be,c^a=cgh,e^a=eg,g^a=gi,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, 389390);