#8115450

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	#392	C₂³
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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

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, 8022646);