#8197196

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	#53698	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4059 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104392);