#390966

order	512
nilpotency class	3
isoclinism	isoclinic class of #232169
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

796 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)(9,10,14,19,15,16,13,29)(11,20,24,30,23,33,21,17)(12,25,27,18,28,34,26,31)(22,32,36,40,38,39,37,35);(1,3,5,2)(4,7,8,6)(10,16)(11,21)(13,14)(17,20)(22,37)(23,24)(25,34)(26,27)(30,33)(32,35)(36,38)(39,40);(1,4,5,8)(2,6,3,7)(9,11,15,23)(10,17)(12,22,28,38)(13,21,14,24)(16,30)(18,32)(19,20)(25,35)(26,37,27,36)(29,33)(31,39)(34,40);(2,3)(6,7)(9,12,15,28)(10,18,16,31)(11,22,23,38)(13,27,14,26)(17,32,30,39)(19,25,29,34)(20,35,33,40)(21,36,24,37)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 298162);