#491427

order	512
nilpotency class	4
isoclinism	isoclinic class of #491239
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

386 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398623);