#382476

order	512
nilpotency class	3
isoclinism	isoclinic class of #184985
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2222 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=h,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,d^c=de,f^c=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, 289672);