#8158800

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3852 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8065996);