#299413

order	512
nilpotency class	3
isoclinism	isoclinic class of #299405
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18624	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

917 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

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, 206609);