#263390

order	512
nilpotency class	3
isoclinism	isoclinic class of #263387
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3109	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

791 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)(3,11)(4,10)(5,9)(6,8)(7,13)(12,16)(14,15)(17,18,22,26)(19,27,30,43)(20,31,34,47)(21,35,36,25)(23,37,38,29)(24,40,42,33)(28,41,32,39)(44,48,46,45);(1,3)(2,7)(4,12)(5,13)(6,14)(8,16)(9,11)(10,15)(17,19,21,29,22,30,36,37)(18,23,25,27,26,38,35,43)(20,32,33,48,34,28,40,45)(24,41,31,46,42,39,47,44);(2,8)(4,6)(7,15)(9,10)(11,16)(12,14)(23,38)(27,43)(39,41)(44,46);(1,4,5,6)(2,8,9,10)(3,12,13,14)(7,16,11,15)(17,20,21,33,22,34,36,40)(18,24,25,31,26,42,35,47)(19,28,29,45,30,32,37,48)(23,39,27,44,38,41,43,46)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,d^2=e,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,f^b=fhi,d^c=di,f^c=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170586);