#8975585

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3775 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,c^2=h,d^2=g,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8882781);