#491212

order	512
nilpotency class	4
isoclinism	isoclinic class of #491091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
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

419 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)(3,11)(4,10)(5,7)(6,9)(8,12)(13,16)(14,15)(17,18,23,27)(19,28,26,29)(20,32,36,42)(21,37,38,25)(22,39,40,31)(24,35,43,33)(30,34,41,48)(44,47,45,46);(2,7)(8,15)(9,10)(11,16)(17,19,29,18,21,31,22,25,23,26,28,27,38,39,40,37)(20,30,45,24,35,47,34,42,36,41,44,43,33,46,48,32);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,11)(10,16)(17,20)(18,24)(19,30)(21,35)(22,34)(23,36)(25,42)(26,41)(27,43)(28,44)(29,45)(31,47)(32,37)(33,38)(39,46)(40,48);(1,4,5,6)(2,9,7,10)(3,12,13,14)(8,11,15,16)(17,21,23,38)(18,25,27,37)(19,31,26,39)(20,33,36,35)(22,28,40,29)(24,32,43,42)(30,46,41,47)(34,45,48,44)

Polycyclic presentation:

2^9|a^2=i,b^2=egh,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^c=di,f^c=fh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398408);