#8741727

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3920 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=h,e^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, 8648923);