#8425501

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4046 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332697);