#8192464

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4080 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8099660);