#8898856

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	#56338	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8806052);