#8504786

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3747 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8411982);