#299416

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

933 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,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, 206612);