#8926546

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	#47129	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3850 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8833742);