#119734

order	512
nilpotency class	3
isoclinism	isoclinic class of #119733
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

550 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)(3,6)(4,7)(5,9)(8,13)(10,14)(11,16)(12,15)(17,18)(19,29)(20,34,38,42)(21,27)(22,26)(23,28)(24,31)(25,36,43,37)(30,45,47,39)(32,40)(33,44)(35,48,46,41);(1,3,5,11)(2,6,9,16)(4,12,13,10)(7,15,8,14)(17,19)(18,24)(20,35)(21,32)(22,31)(23,33)(25,41)(26,44)(27,29)(28,40)(30,37)(34,45)(36,47)(38,46)(39,42)(43,48);(2,7)(4,13)(6,14)(8,9)(10,12)(15,16)(17,20,21,36,23,38,22,37)(18,25,26,42,28,43,27,34)(19,30,31,46,33,47,32,35)(24,39,29,41,40,45,44,48)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=h,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=dh,e^b=ei,f^b=fh,e^c=ei,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, 26930);