#8187010

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

4021 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=g,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

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, 8094206);