#8095796

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3979 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=hi,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002992);