#10312372

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	#53063	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3991 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,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, 10219568);