#485017

order	512
nilpotency class	4
isoclinism	isoclinic class of #484199
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	#61000	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#433 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

732 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)(17,18,23,28)(19,25,31,45)(20,26,22,38)(21,34,37,48)(24,39,42,30)(27,33,46,36)(29,43,32,44)(35,40,47,41);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16)(17,19)(18,24)(20,29)(21,35)(22,32)(23,31)(25,34)(26,40)(27,44)(28,42)(30,36)(33,39)(37,47)(38,41)(43,46)(45,48);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16)(17,19)(18,25)(20,32)(21,30)(22,29)(23,31)(24,34)(26,44)(27,40)(28,45)(33,47)(35,36)(37,39)(38,43)(41,46)(42,48);(1,4,5,11)(2,7,8,14)(3,9,10,15)(6,13,12,16)(17,20,23,22)(18,26,28,38)(19,29,31,32)(21,33,37,36)(24,40,42,41)(25,43,45,44)(27,34,46,48)(30,35,39,47)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392213);