#479876

order	512
nilpotency class	3
isoclinism	isoclinic class of #476707
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

620 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)(3,8)(4,6)(5,7)(9,10,14,19)(11,20)(12,25)(13,15,30,18)(16,24)(17,26)(21,34,38,40)(22,33)(23,31)(27,32,39,37)(28,35)(29,36);(1,3,5,4)(2,6,7,8)(10,15)(12,26)(13,30)(16,31)(18,19)(20,33)(21,27)(22,23)(25,36)(28,29)(34,40)(38,39);(1,4,5,3)(2,6,7,8)(9,11,13,22,14,24,30,23)(10,16,15,31,19,20,18,33)(12,27,26,38,29,39,28,21)(17,34,36,32,35,40,25,37);(3,4)(6,8)(9,12,14,29)(10,17)(11,21,24,38)(13,28,30,26)(15,25)(16,32)(18,36)(19,35)(20,37)(22,39,23,27)(31,34)(33,40)

Polycyclic presentation:

2^9|a^2=g,b^2=efgh,c^2=ef,d^2=fi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=def,e^b=ei,h^b=hi,d^c=de,g^c=gi,h^c=hi,e^d=ei,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387072);