#8341619

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	#43969	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3998 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=h,e^2=hi,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, 8248815);