#8112553

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	#56210	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3854 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:

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

Polycyclic presentation:

2^9|c^2=gh,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8019749);