#8839081

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

4129 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)(5,13,14,9)(11,12,15,16)(17,18,22,24)(19,23,25,20)(21,29,31,26)(27,28,30,32)(33,34)(35,39)(36,40)(37,46)(38,42)(41,47)(43,44)(45,48);(2,7)(4,10)(5,14)(11,15)(17,19)(18,23)(20,24)(21,30)(22,25)(26,28)(27,31)(29,32)(33,35)(34,39)(36,37)(38,43)(40,46)(41,48)(42,44)(45,47);(2,7)(3,8)(9,13)(11,15)(18,24)(19,25)(26,29)(27,30)(35,43)(36,45)(39,44)(40,48);(1,3)(2,4)(5,11)(6,8)(7,10)(9,12)(13,16)(14,15)(17,19)(18,23)(20,24)(21,27)(22,25)(26,32)(28,29)(30,31)(33,36,38,45)(34,40,42,48)(35,37,43,47)(39,46,44,41);(1,4)(2,8)(3,7)(5,12)(6,10)(9,15)(11,13)(14,16)(17,20,22,23)(18,25,24,19)(21,28,31,32)(26,30,29,27)(33,37)(34,41)(35,36)(38,47)(39,48)(40,44)(42,46)(43,45);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,21,22,31)(18,26,24,29)(19,27,25,30)(20,28,23,32)(33,34)(35,44)(36,40)(37,41)(38,42)(39,43)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=h,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, 8746277);