#8260653

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	#47074	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4182 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18,22,23)(19,27,28,24)(20,21,26,25)(29,31,32,30)(33,34,39,42)(35,40,46,41)(36,47,37,38)(43,45,44,48);(2,6)(7,9)(8,13)(14,15)(18,23)(19,28)(20,26)(30,31)(33,35,39,46)(34,40,42,41)(36,44,37,43)(38,45,47,48);(4,12)(8,13)(10,16)(14,15)(18,23)(21,25)(24,27)(30,31)(33,36,39,37)(34,38,42,47)(35,43,46,44)(40,48,41,45);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,19)(18,24)(20,29)(21,30)(22,28)(23,27)(25,31)(26,32)(33,37,39,36)(34,38,42,47)(35,44,46,43)(40,48,41,45);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,22,26)(18,25,23,21)(19,29,28,32)(24,31,27,30)(33,35)(34,41)(36,43)(37,44)(38,45)(39,46)(40,42)(47,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,25)(18,26,23,20)(19,30,28,31)(24,32,27,29)(33,38)(34,36)(35,45)(37,42)(39,47)(40,44)(41,43)(46,48)

Polycyclic presentation:

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

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, 8167849);