#482490

order	512
nilpotency class	4
isoclinism	isoclinic class of #482362
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	C₂²×#41	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

863 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)(19,29,33,40)(20,34)(21,27,22,26)(23,28)(24,32)(25,38)(30,46,35,45)(31,41)(36,43,37,44)(39,47)(42,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19,23,33)(18,24,28,41)(20,35,38,30)(21,32,22,31)(25,42,34,39)(26,29,27,40)(36,47,37,48)(43,46,44,45);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,23,38)(18,25,28,34)(19,30,33,35)(21,36,22,37)(24,39,41,42)(26,43,27,44)(29,45,40,46)(31,47,32,48);(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,36,38,37)(24,40,41,29)(25,43,34,44)(30,47,35,48)(39,46,42,45)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389686);