#290055

order	512
nilpotency class	3
isoclinism	isoclinic class of #290011
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

615 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:

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

Polycyclic presentation:

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

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, 197251);