#8464038

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

4151 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)(18,23)(19,26)(25,30)(27,32)(34,39)(36,40)(42,45)(44,47);(3,12)(7,8)(13,16)(14,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,31)(28,32)(29,30)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(1,3)(2,7)(4,13)(5,14)(6,12)(8,11)(9,15)(10,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,30)(29,32)(33,35,38,41)(34,40,39,36)(37,43,46,48)(42,47,45,44);(1,3)(2,8)(4,13)(5,15)(6,12)(7,11)(9,14)(10,16)(17,19)(18,21)(20,27)(22,26)(23,24)(25,28)(29,32)(30,31)(33,34)(35,40)(36,41)(37,45)(38,39)(42,46)(43,44)(47,48);(1,4,6,10)(2,9,11,5)(3,13,12,16)(7,15,8,14)(17,20,22,29)(18,25,23,30)(19,27,26,32)(21,28,24,31)(33,36,38,40)(34,41,39,35)(37,44,46,47)(42,48,45,43);(1,5,6,9)(2,10,11,4)(3,14,12,15)(7,16,8,13)(17,21)(18,26)(19,23)(20,28)(22,24)(25,32)(27,30)(29,31)(33,37)(34,42)(35,43)(36,44)(38,46)(39,45)(40,47)(41,48)

Polycyclic presentation:

2^9|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, 8371234);