#8831292

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	#56486	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4016 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8738488);