#115239

order	512
nilpotency class	3
isoclinism	isoclinic class of #9930
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#3006	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

869 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,f^a=fi,c^b=cf,d^b=dh,f^b=fh,d^c=di,e^c=eh,f^c=fh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 22435);