#8757412

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3701 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)(3,7)(4,10)(5,8)(6,14)(9,12)(11,13)(15,16)(17,18,23,28)(19,27,22,24)(20,21,26,25)(29,32,31,30)(35,37)(39,41)(43,45)(47,48);(1,3)(2,9)(4,11)(5,7)(6,15)(8,12)(10,16)(13,14)(17,19,23,22)(18,24,28,27)(20,31,26,29)(21,32,25,30)(33,34,38,42)(35,39,37,41)(36,44,46,40)(43,48,45,47);(1,4,8,6)(2,10,5,14)(3,11,12,15)(7,13,9,16)(17,20,23,26)(18,25,28,21)(19,29,22,31)(24,32,27,30)(33,35,38,37)(34,39,42,41)(36,43,46,45)(40,47,44,48);(1,5,8,2)(3,7,12,9)(4,10,6,14)(11,16,15,13)(17,19,23,22)(18,24,28,27)(20,29,26,31)(21,30,25,32);(1,6,8,4)(2,10,5,14)(3,11,12,15)(7,16,9,13)(17,21,23,25)(18,26,28,20)(19,30,22,32)(24,31,27,29)(33,36,38,46)(34,40,42,44)(35,43,37,45)(39,47,41,48);(1,7)(2,3)(4,13)(5,12)(6,16)(8,9)(10,11)(14,15)(17,22,23,19)(18,27,28,24)(20,31,26,29)(21,32,25,30)(33,37,38,35)(34,41,42,39)(36,45,46,43)(40,48,44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=g,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8664608);