#8157148

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
prime radical	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3884 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=hi,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8064344);