#8987295

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	#54025	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4061 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=ghi,e^2=g,f^2=gh,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, 8894491);