#8661369

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	#41301	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,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, 8568565);