#10055417

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	#41158	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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)(3,7)(4,8)(5,6)(9,13)(10,15)(11,16)(12,14)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30)(33,34)(35,40)(36,45)(37,42)(38,47)(39,41)(43,44)(46,48);(2,6)(7,13)(8,14)(15,16)(18,20)(19,25)(24,28)(27,31)(33,35)(34,40)(36,46)(37,39)(38,44)(41,42)(43,47)(45,48);(3,9)(7,13)(10,11)(15,16)(18,20)(19,25)(24,28)(27,31)(34,41)(35,37)(36,47)(44,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,15)(11,16)(12,14)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,36)(34,38)(35,43)(37,46)(39,47)(40,48)(41,45)(42,44);(1,3,5,9)(2,7,6,13)(4,11,12,10)(8,16,14,15)(17,20,22,18)(19,26,25,23)(21,24,29,28)(27,30,31,32)(33,37,39,35)(34,42,41,40)(36,46,47,43)(38,44,45,48);(1,4,5,12)(2,8,6,14)(3,10,9,11)(7,15,13,16)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,38)(34,36)(35,44)(37,48)(39,45)(40,46)(41,47)(42,43)

Polycyclic presentation:

2^9|a^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,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, 9962613);