#8093465

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	#52643	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3859 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000661);