#8202201

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	#56791	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4004 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=h,f^2=g,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, 8109397);