#8095386

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	#53101	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4035 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=hi,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8002582);