#8912285

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	#50599	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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)(3,7,10,4)(5,8,14,16)(11,13,15,12)(17,18)(19,23)(20,25)(21,22)(24,30)(26,32)(27,31)(28,29)(33,34)(35,41)(36,42)(37,40)(38,39)(43,47)(44,48)(45,46);(1,3)(2,7)(4,9)(5,11)(6,10)(8,13)(12,16)(14,15)(17,19,22,29)(18,23,21,28)(20,27,30,26)(24,32,25,31)(34,39)(35,42)(37,45)(44,47);(3,10)(4,7)(11,15)(12,13)(17,20,22,30)(18,24,21,25)(19,26,29,27)(23,31,28,32)(33,35)(34,36)(37,43)(38,42)(39,41)(40,44)(45,48)(46,47);(1,4,6,7)(2,3,9,10)(5,13,14,12)(8,15,16,11)(17,20)(18,25)(19,27)(21,24)(22,30)(23,31)(26,29)(28,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48);(1,5)(2,8)(3,11)(4,12)(6,14)(7,13)(9,16)(10,15)(17,19)(18,23)(20,26)(21,28)(22,29)(24,31)(25,32)(27,30)(33,37,38,45)(34,40,39,46)(35,43,42,48)(36,44,41,47);(1,3)(2,7)(4,9)(5,11)(6,10)(8,13)(12,16)(14,15)(17,21,22,18)(19,28,29,23)(20,24,30,25)(26,31,27,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=hi,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,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, 8819481);