#8944507

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	#35827	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4007 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)(3,5)(4,13)(6,14)(7,9)(8,12)(10,11)(15,16)(17,18,23,22)(19,25,20,24)(21,26,30,29)(27,32,28,31)(33,34,38,41)(35,39,45,48)(36,40,46,37)(42,43,44,47);(1,3,7,12)(2,8,9,5)(4,10,14,16)(6,11,13,15)(17,19,23,20)(18,24,22,25)(21,27,30,28)(26,31,29,32)(33,35)(34,39)(36,44)(37,43)(38,45)(40,47)(41,48)(42,46);(2,9)(5,8)(6,13)(11,15)(17,19,23,20)(18,25,22,24)(21,28,30,27)(26,31,29,32)(33,36,38,46)(34,40,41,37)(35,42,45,44)(39,43,48,47);(1,4)(2,6)(3,10)(5,15)(7,14)(8,11)(9,13)(12,16)(17,20,23,19)(18,25,22,24)(21,27,30,28)(26,31,29,32)(33,37)(34,36)(35,43)(38,40)(39,44)(41,46)(42,48)(45,47);(1,5,7,8)(2,3,9,12)(4,11,14,15)(6,16,13,10)(17,21,23,30)(18,26,22,29)(19,27,20,28)(24,31,25,32)(33,36)(34,40)(35,44)(37,41)(38,46)(39,47)(42,45)(43,48);(1,6)(2,4)(3,11)(5,16)(7,13)(8,10)(9,14)(12,15)(17,22,23,18)(19,25,20,24)(21,29,30,26)(27,32,28,31)(33,35)(34,39)(36,44)(37,43)(38,45)(40,47)(41,48)(42,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8851703);