#8898574

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4138 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

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, 8805770);