#8734609

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	#52839	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3960 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8641805);