#10480822

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	#55437	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4190 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)(3,10)(4,7)(5,8)(6,12)(9,14)(11,15)(13,16)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,34)(35,42)(36,41)(37,40)(38,39)(43,48)(44,47)(45,46);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(18,23)(24,25)(26,30)(31,32)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(1,2,7,4)(3,10,5,8)(6,9,14,12)(11,16,13,15)(17,19,22,20)(18,24,23,25)(21,27,29,28)(26,31,30,32)(33,35,39,37)(34,40,38,42)(36,43,46,44)(41,47,45,48);(1,4,7,2)(3,10,5,8)(6,12,14,9)(11,16,13,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,36)(34,41)(35,43)(37,44)(38,45)(39,46)(40,47)(42,48);(1,5,7,3)(2,8,4,10)(6,13,14,11)(9,15,12,16)(17,20,22,19)(18,25,23,24)(21,27,29,28)(26,31,30,32)(33,37,39,35)(34,40,38,42)(36,43,46,44)(41,48,45,47);(1,6,7,14)(2,9,4,12)(3,11,5,13)(8,15,10,16)(17,21,22,29)(18,26,23,30)(19,27,20,28)(24,31,25,32)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=i,e^2=g,f^2=g,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, 10388018);