#8581795

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

4288 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)(3,10,8,4)(5,9,14,15)(11,16,13,12)(18,19)(23,25)(24,26)(30,31)(33,34,39,36)(35,37,40,42)(38,41,46,44)(43,45,47,48);(2,7)(3,8)(9,15)(11,13)(17,18,22,19)(20,23,27,25)(21,24,29,26)(28,30,32,31)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,48)(46,47);(2,7)(4,10)(9,15)(12,16)(20,27)(23,25)(28,32)(30,31)(33,34,39,36)(35,42,40,37)(38,41,46,44)(43,48,47,45);(1,3,6,8)(2,4,7,10)(5,13,14,11)(9,16,15,12)(17,19,22,18)(20,25,27,23)(21,24,29,26)(28,30,32,31)(33,36,39,34)(35,42,40,37)(38,41,46,44)(43,45,47,48);(1,4)(2,8)(3,7)(5,12)(6,10)(9,13)(11,15)(14,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,37,39,42)(34,40,36,35)(38,45,46,48)(41,47,44,43);(1,5,6,14)(2,9,7,15)(3,11,8,13)(4,12,10,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,46)(34,41,36,44)(35,43,40,47)(37,45,42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=gi,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8488991);