#8202284

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	#56544	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=h,e^2=g,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, 8109480);