#121821

order	512
nilpotency class	3
isoclinism	isoclinic class of #10645
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#397
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1445
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

383 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,11)(4,9)(5,8)(6,10)(7,12)(13,15)(14,16)(17,18,23,26,22,27,21,36)(19,28,32,39,31,40,30,24)(20,25,34,43,35,44,37,42)(29,41,45,48,46,47,33,38);(1,3)(2,7)(4,13)(5,12)(6,14)(8,16)(9,11)(10,15)(17,19,23,32,22,31,21,30)(18,24,36,40,27,39,26,28)(20,33,34,29,35,45,37,46)(25,41,42,38,44,47,43,48);(2,8)(3,12)(4,5)(7,15)(9,10)(13,14)(17,20,23,34,22,35,21,37)(18,25,36,42,27,44,26,43)(19,29,32,45,31,46,30,33)(24,38,40,47,39,48,28,41)

Polycyclic presentation:

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

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, 29017);