#8093316

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	#45592	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4056 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18,23,24)(19,25,29,32)(20,30,31,21)(22,28,27,26)(33,34,39,42)(35,38,45,40)(36,41,46,37)(43,44,47,48);(2,6)(3,10)(8,12)(11,16)(18,24)(21,30)(25,32)(26,28)(36,46)(37,41)(43,47)(44,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,19,23,29)(18,25,24,32)(20,27,31,22)(21,28,30,26)(33,35,39,45)(34,40,42,38)(36,43,46,47)(37,44,41,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,20)(18,21)(19,27)(22,29)(23,31)(24,30)(25,28)(26,32)(33,36)(34,41)(35,43)(37,42)(38,44)(39,46)(40,48)(45,47);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,21)(18,20)(19,28)(22,32)(23,30)(24,31)(25,27)(26,29)(33,37,39,41)(34,36,42,46)(35,44,45,48)(38,47,40,43);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22,23,27)(18,26,24,28)(19,20,29,31)(21,32,30,25)(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=gh,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, 8000512);