#8942400

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

4157 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8849596);