#481836

order	512
nilpotency class	4
isoclinism	isoclinic class of #481819
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	#34633	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

1075 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^2=h,f^2=i,b^a=be,c^a=ch,e^a=eg,g^a=gi,d^b=di,e^b=eh,f^b=fgh,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, 389032);