#116376

order	512
nilpotency class	3
isoclinism	isoclinic class of #116371
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

683 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)(3,7)(4,6)(5,12)(8,11)(9,15)(10,14)(13,16)(17,18,21,24,22,26,20,25)(19,23,27,31,28,32,29,30);(2,6)(3,9,10,13)(7,15,14,16)(8,12)(17,19)(18,23)(20,29)(21,27)(22,28)(24,31)(25,30)(26,32);(1,3)(2,7)(4,10)(5,13)(6,14)(8,15)(9,11)(12,16)(18,24)(20,21)(23,30)(25,26)(27,29)(31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=dh,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,e^b=ei,d^c=dh,e^c=ei,f^c=fh,g^c=gi

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, 23572);