#317971

order	512
nilpotency class	3
isoclinism	isoclinic class of #317966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2537
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3253	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

521 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=egi,d^2=e,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=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, 225167);