#10320377

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	#49741	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3967 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)(3,6)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(18,22)(20,28)(23,25)(26,32)(33,34,39,37)(35,36,40,44)(38,41,47,45)(42,43,48,46);(3,9)(6,13)(10,15)(14,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(33,35,39,40)(34,36,37,44)(38,42,47,48)(41,43,45,46);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,18,21,22)(19,23,27,25)(20,24,28,29)(26,30,32,31)(33,36)(34,35)(37,40)(38,43)(39,44)(41,42)(45,48)(46,47);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,37,39,34)(35,36,40,44)(38,45,47,41)(42,43,48,46);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36)(34,40)(35,37)(38,46)(39,44)(41,42)(43,47)(45,48);(1,4,5,12)(2,7,8,11)(3,10,9,15)(6,14,13,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,38,39,47)(34,41,37,45)(35,42,40,48)(36,43,44,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=ghi,d^2=gi,e^2=g,f^2=h,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, 10227573);