#8108015

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3921 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:

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

Polycyclic presentation:

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

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, 8015211);