#8202195

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	#40696	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4226 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=gh,e^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, 8109391);