#285098

order	512
nilpotency class	3
isoclinism	isoclinic class of #285051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3508	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

681 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=hi,d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=di,e^b=eh,g^b=gh,f^c=fi,g^c=ghi,e^d=eh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192294);