#8436654

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

4035 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=ghi,d^2=gh,e^2=g,f^2=g,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, 8343850);