#8691726

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	#51857	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4131 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8598922);