#115561

order	512
nilpotency class	3
isoclinism	isoclinic class of #10120
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1438
prime radical	#60702	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

798 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)(3,10,12,6)(4,9)(5,8)(7,11,16,13)(14,15)(17,18,22,27,23,28,21,26)(19,24,31,39,30,38,35,43)(20,25,34,42,33,41,36,44)(29,45,47,40,46,37,32,48);(1,3)(2,6)(4,13)(5,12)(7,15)(8,10)(9,16)(11,14)(17,19)(18,24)(20,32)(21,35)(22,31)(23,30)(25,40)(26,43)(27,39)(28,38)(29,34)(33,47)(36,46)(37,42)(41,48)(44,45);(1,3)(2,7)(4,13)(5,12)(6,9)(8,16)(10,15)(11,14)(17,20)(18,25)(19,29)(21,36)(22,34)(23,33)(24,37)(26,44)(27,42)(28,41)(30,46)(31,47)(32,35)(38,45)(39,48)(40,43)

Polycyclic presentation:

2^9|a^2=g,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fh,c^b=cf,e^b=ehi,f^b=fi,g^b=gh,d^c=di,e^c=eh,f^c=fi,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, 22757);