#8425417

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

3976 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=gh,f^2=g,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, 8332613);