#10195495

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	#58238	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=h,e^2=g,f^2=hi,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, 10102691);