#8946211

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	#50053	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=ghi,d^2=g,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8853407);