#8905245

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

4100 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8812441);