#8157392

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	#58007	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4011 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:

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

Polycyclic presentation:

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

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, 8064588);