#8228349

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	#52308	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4306 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)(17,18,22,24)(19,27,29,23)(20,30,31,25)(21,26,32,28)(33,34,38,40)(35,39,43,48)(36,41,45,46)(37,44,47,42);(2,6)(7,9)(8,13)(14,15)(17,19,22,29)(18,23,24,27)(20,26,31,28)(21,25,32,30)(33,35)(34,39)(36,44)(37,46)(38,43)(40,48)(41,47)(42,45);(4,12)(8,13)(10,16)(14,15)(18,24)(21,32)(23,27)(25,30)(34,40)(35,43)(41,46)(42,44);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,19)(18,23)(20,28)(21,25)(22,29)(24,27)(26,31)(30,32)(33,36)(34,41)(35,42)(37,39)(38,45)(40,46)(43,44)(47,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,20,22,31)(18,25,24,30)(19,28,29,26)(21,27,32,23)(33,37)(34,42)(35,41)(36,39)(38,47)(40,44)(43,46)(45,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,21)(18,26)(19,25)(20,27)(22,32)(23,31)(24,28)(29,30)(33,35)(34,39)(36,42)(37,41)(38,43)(40,48)(44,45)(46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=ghi,e^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, 8135545);