#8324619

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	#40036	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3989 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=gh,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

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, 8231815);