#10480816

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	#55320	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4024 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)(4,8)(5,7)(6,9)(11,15)(12,14)(13,16)(17,18)(19,20)(21,29)(22,23)(24,27)(25,31)(26,32)(28,30)(36,44)(39,45)(42,48)(46,47);(1,3,6,5)(2,7,9,10)(4,12,13,11)(8,15,16,14)(18,23)(20,24)(25,29)(28,32)(33,34)(35,40)(36,45)(37,41)(38,43)(39,44)(42,46)(47,48);(17,19)(18,24)(20,23)(21,26)(22,27)(25,32)(28,29)(30,31)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,48)(45,47);(1,4,6,13)(2,8,9,16)(3,11,5,12)(7,14,10,15)(17,20,22,24)(18,19,23,27)(21,28,31,32)(25,26,29,30)(33,36,37,44)(34,39,41,45)(35,42,43,48)(38,46,40,47);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,19)(18,20)(21,30)(22,27)(23,24)(25,32)(26,31)(28,29)(33,35)(34,40)(36,42)(37,43)(38,41)(39,47)(44,48)(45,46);(1,5,6,3)(2,7,9,10)(4,12,13,11)(8,14,16,15)(17,21,22,31)(18,25,23,29)(19,26,27,30)(20,28,24,32)(33,34)(35,38)(36,39)(37,41)(40,43)(42,46)(44,45)(47,48)

Polycyclic presentation:

2^9|b^2=gh,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, 10388012);