#8201987

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	#36758	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4234 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^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, 8109183);