#8843371

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

4285 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:

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

Polycyclic presentation:

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

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, 8750567);