#121121

order	512
nilpotency class	3
isoclinism	isoclinic class of #121121
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1443
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^c=fi,g^c=ghi

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, 28317);