#8755045

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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)(3,8)(4,6)(5,14)(7,9)(10,13)(11,16)(12,15)(18,22)(20,27)(23,28)(26,31)(33,34,39,42)(35,43,46,37)(36,41,38,40)(44,47,45,48);(1,3)(2,8)(4,13)(5,11)(6,10)(7,12)(9,15)(14,16)(19,25)(23,28)(26,31)(30,32)(33,35,39,46)(34,37,42,43)(36,44,38,45)(40,47,41,48);(2,9)(3,12)(4,11)(10,14)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32)(33,35)(34,37)(36,44)(38,45)(39,46)(40,47)(41,48)(42,43);(1,4,7,11)(2,6,9,16)(3,13,12,5)(8,10,15,14)(33,36,39,38)(34,40,42,41)(35,44,46,45)(37,47,43,48);(1,5,7,13)(2,10,9,14)(3,11,12,4)(6,15,16,8)(17,19,21,25)(18,23,22,28)(20,26,27,31)(24,30,29,32)(33,37)(34,35)(36,47)(38,48)(39,43)(40,44)(41,45)(42,46);(1,6)(2,11)(3,14)(4,9)(5,15)(7,16)(8,13)(10,12)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)(33,38,39,36)(34,41,42,40)(35,45,46,44)(37,48,43,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,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, 8662241);