#8147849

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

4138 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=gi,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8055045);