#10171908

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	#38331	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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:

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

Polycyclic presentation:

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

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, 10079104);