#8505410

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	#48051	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4109 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=i,d^2=i,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8412606);