#10407817

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4022 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,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, 10315013);