#10480818

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	#55417	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4018 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=g,d^2=gi,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, 10388014);