#10407856

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	#38540	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4277 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=hi,d^2=hi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10315052);