#115527

order	512
nilpotency class	3
isoclinism	isoclinic class of #115519
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#3213	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

969 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=h,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=dh,f^b=fhi,d^c=di,f^c=fhi,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, 22723);