#10171838

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	#49217	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=h,d^2=hi,e^2=gi,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, 10079034);