#482394

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	C₂×#532	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1009 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)(9,10,13,18,14,17,28,34)(11,19,22,35,20,15,29,31)(12,23,25,36,26,16,27,33)(21,30,37,39,24,32,38,40);(2,5)(3,8)(6,7)(9,11)(10,15)(12,24)(13,22)(14,20)(16,32)(17,19)(18,35)(21,26)(23,30)(25,38)(27,37)(28,29)(31,34)(33,39)(36,40);(2,6)(5,7)(11,20)(15,19)(21,24)(22,29)(30,32)(31,35)(37,38)(39,40);(2,6)(5,7)(9,12,14,26)(10,16,17,23)(11,21,20,24)(13,27,28,25)(15,30,19,32)(18,36,34,33)(22,38,29,37)(31,40,35,39)

Polycyclic presentation:

2^9|a^2=f,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, 389590);