#490042

order	512
nilpotency class	4
isoclinism	isoclinic class of #489719
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	C₄×C₈	C₂⁴
prime radical	#121	D₈
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2966 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

661 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397238);