#482396

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×C₄⋊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×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

997 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=i,e^2=h,f^2=i,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, 389592);