#8183851

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

3997 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)(3,4)(5,8)(6,14)(7,10)(9,15)(11,13)(12,16)(18,20)(19,25)(24,28)(27,31)(33,34,38,39)(35,37,42,40)(36,44,46,41)(43,47,48,45);(1,3,7,8)(2,4,10,5)(6,11,15,16)(9,12,14,13)(17,18,22,20)(19,23,25,26)(21,24,29,28)(27,30,31,32)(34,39)(35,42)(41,44)(43,48);(1,4,7,5)(2,8,10,3)(6,12,15,13)(9,16,14,11)(17,18,22,20)(19,23,25,26)(21,28,29,24)(27,32,31,30)(33,35,38,42)(34,40,39,37)(36,43,46,48)(41,47,44,45);(1,5,7,4)(2,8,10,3)(6,13,15,12)(9,16,14,11)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,32)(29,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,6,7,15)(2,9,10,14)(3,11,8,16)(4,12,5,13)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,2)(3,4)(5,8)(6,9)(7,10)(11,12)(13,16)(14,15)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,37,38,40)(34,42,39,35)(36,45,46,47)(41,48,44,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=h,d^2=gh,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, 8091047);