#482639

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

900 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=hi,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,c^b=ci,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, 389835);