#488706

order	512
nilpotency class	4
isoclinism	isoclinic class of #488699
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	#62258	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#435 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

743 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 395902);