#8197116

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4041 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=i,d^2=ghi,e^2=h,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, 8104312);