#8095267

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	#45712	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3977 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=i,d^2=gh,e^2=i,f^2=g,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, 8002463);