#317984

order	512
nilpotency class	3
isoclinism	isoclinic class of #317982
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	#19332	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

532 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:

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

Polycyclic presentation:

2^9|b^2=eg,d^2=eh,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=di,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, 225180);