#8093314

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	#45609	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000510);