#482162

order	512
nilpotency class	4
isoclinism	isoclinic class of #481601
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

913 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)(19,29)(20,34,36,25)(21,26)(22,38,39,27)(23,28)(24,32,42,46)(30,44,47,41)(31,40)(33,45)(35,43)(37,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,23,33)(18,24,28,42)(20,30,36,47)(21,37,35,31)(22,32,39,46)(25,40,34,48)(26,44,43,41)(27,29,38,45);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,23,36)(18,25,28,34)(19,30,33,47)(21,22,35,39)(24,40,42,48)(26,27,43,38)(29,41,45,44)(31,46,37,32);(17,21,23,35)(18,26,28,43)(19,31,33,37)(20,22,36,39)(24,41,42,44)(25,27,34,38)(29,40,45,48)(30,46,47,32)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=i,e^2=h,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, 389358);