#121762

order	512
nilpotency class	3
isoclinism	isoclinic class of #112870
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#390
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1445
prime radical	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

427 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:

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

Polycyclic presentation:

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

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, 28958);