#121219

order	512
nilpotency class	3
isoclinism	isoclinic class of #117993
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#388
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1443
prime radical	C₂⁴	#126
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

367 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)(4,7)(5,9)(6,10)(8,14)(12,15)(13,16)(17,18)(19,28,31,24)(20,32)(21,36)(22,27)(23,26)(25,34)(29,46,48,42)(30,40,37,41)(33,39,47,45)(35,44)(38,43);(2,7)(3,12)(4,5)(8,11)(9,10)(15,16)(17,19,21,30,23,31,22,37)(18,24,36,41,26,28,27,40)(20,33,34,29,35,47,38,48)(25,42,44,39,43,46,32,45);(1,3,4,13,6,12,5,14)(2,8,9,15,10,16,7,11)(17,20)(18,25,26,43)(19,29)(21,34)(22,38)(23,35)(24,39,28,45)(27,32,36,44)(30,47)(31,48)(33,37)(40,42,41,46)

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=d,d^2=i,e^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fhi,g^b=ghi,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, 28415);