#485024

order	512
nilpotency class	4
isoclinism	isoclinic class of #484212
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#12110	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2922 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

699 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=i,d^2=h,e^2=gi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fi,g^a=gi,e^b=eg,f^b=fh,g^b=gi,e^c=eh,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392220);