#8976952

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

4083 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)(19,24)(20,23)(21,25)(22,26)(27,29)(28,31)(30,32)(33,34,37,41)(35,40,42,38)(36,39,45,48)(43,47,44,46);(1,2,6,7)(3,8,11,16)(4,9,14,13)(5,10,15,12)(17,19)(18,20)(21,27)(22,23)(24,26)(25,28)(29,32)(30,31)(35,42)(38,40)(43,44)(46,47);(2,7)(3,11)(4,14)(5,15)(17,20,22,24)(18,23,26,19)(21,29,30,28)(25,27,32,31)(33,35,37,42)(34,38,41,40)(36,44,45,43)(39,47,48,46);(1,3)(2,8)(4,10)(5,13)(6,11)(7,16)(9,15)(12,14)(17,20)(18,19)(21,28)(22,24)(23,26)(25,27)(29,30)(31,32)(33,36,37,45)(34,39,41,48)(35,43,42,44)(38,46,40,47);(1,4)(2,9)(3,12)(5,8)(6,14)(7,13)(10,11)(15,16)(17,19)(18,24)(20,26)(21,27)(22,23)(25,29)(28,32)(30,31)(33,35)(34,40)(36,44)(37,42)(38,41)(39,46)(43,45)(47,48);(1,5)(2,10)(3,13)(4,8)(6,15)(7,12)(9,11)(14,16)(17,21,22,30)(18,25,26,32)(19,27,23,31)(20,28,24,29)(33,34)(35,40)(36,39)(37,41)(38,42)(43,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gh,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8884148);