#121867

order	512
nilpotency class	3
isoclinism	isoclinic class of #121865
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

403 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=hi,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=di,e^b=ei,f^b=fhi,g^b=gi,d^c=dh,e^c=eh,f^c=fhi

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, 29063);