#8464041

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

4153 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)(3,8)(4,9)(5,10)(6,11)(7,12)(13,14)(15,16)(17,18,22,26)(19,27,29,23)(20,24,30,32)(21,25,31,28)(33,34,39,42)(35,40,37,36)(38,41,47,46)(43,48,45,44);(3,12)(7,8)(13,16)(14,15)(17,19,22,29)(18,23,26,27)(20,28,30,25)(21,24,31,32)(33,35,39,37)(34,36,42,40)(38,43,47,45)(41,44,46,48);(1,3)(2,7)(4,13)(5,14)(6,12)(8,11)(9,15)(10,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,31)(30,32)(33,36,39,40)(34,35,42,37)(38,44,47,48)(41,43,46,45);(1,3)(2,8)(4,13)(5,15)(6,12)(7,11)(9,14)(10,16)(17,18)(19,27)(20,24)(21,28)(22,26)(23,29)(25,31)(30,32)(33,34)(35,36)(37,40)(38,46)(39,42)(41,47)(43,48)(44,45);(1,4,6,10)(2,9,11,5)(3,13,12,16)(7,15,8,14)(17,20,22,30)(18,24,26,32)(19,28,29,25)(21,23,31,27)(33,37,39,35)(34,40,42,36)(38,45,47,43)(41,48,46,44);(1,5,6,9)(2,10,11,4)(3,14,12,15)(7,16,8,13)(17,21)(18,25)(19,24)(20,23)(22,31)(26,28)(27,30)(29,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8371237);