#8581512

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	#53570	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4030 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8488708);