#391758

order	512
nilpotency class	3
isoclinism	isoclinic class of #213798
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1093 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:

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

Polycyclic presentation:

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

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, 298954);