#8430530

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

4166 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8337726);