#10480795

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	#45917	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4080 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)(3,6)(4,13)(5,9)(7,11)(8,10)(12,16)(14,15)(20,28)(23,29)(26,32)(30,31)(33,34)(35,44)(36,40)(37,41)(38,42)(39,43)(45,47)(46,48);(1,3)(2,8)(4,12)(5,15)(6,11)(7,10)(9,16)(13,14)(17,18)(19,24)(20,29)(21,25)(22,27)(23,28)(26,30)(31,32)(33,35,39,42)(34,38,43,44)(36,45,41,48)(37,47,40,46);(1,4,7,5)(2,9,11,13)(3,12,10,15)(6,14,8,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,32)(29,31)(33,36,39,41)(34,40,43,37)(35,45,42,48)(38,46,44,47);(1,5,7,4)(2,9,11,13)(3,12,10,15)(6,16,8,14)(17,20,21,28)(18,23,25,29)(19,26,27,32)(22,30,24,31)(33,37,39,40)(34,41,43,36)(35,46,42,47)(38,45,44,48);(1,3)(2,6)(4,12)(5,15)(7,10)(8,11)(9,14)(13,16)(17,19)(18,24)(20,26)(21,27)(22,25)(23,31)(28,32)(29,30)(33,34)(35,38)(36,40)(37,41)(39,43)(42,44)(45,46)(47,48);(1,6)(2,10)(3,11)(4,14)(5,16)(7,8)(9,15)(12,13)(17,18)(19,22)(20,23)(21,25)(24,27)(26,30)(28,29)(31,32)(33,38)(34,42)(35,43)(36,46)(37,45)(39,44)(40,48)(41,47)

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10387991);