#8463761

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3957 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)(3,12)(4,9)(5,15)(6,16)(7,11)(8,14)(10,13)(18,20)(23,26)(24,28)(30,32)(33,34,39,38)(35,42,37,40)(36,41,46,45)(43,48,44,47);(1,3)(2,8)(4,6)(5,13)(7,14)(9,10)(11,12)(15,16)(19,25)(23,26)(27,31)(30,32)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(1,4,7,5)(2,9,11,15)(3,6,14,13)(8,10,12,16)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,5,7,4)(2,9,11,15)(3,13,14,6)(8,10,12,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,37,39,35)(34,42,38,40)(36,44,46,43)(41,48,45,47);(1,6,7,13)(2,10,11,16)(3,4,14,5)(8,9,12,15)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30);(1,2)(3,8)(4,9)(5,15)(6,10)(7,11)(12,14)(13,16)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gh,d^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8370957);