#8472850

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	#45529	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4180 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)(3,9)(4,8)(5,16)(6,10)(7,11)(12,15)(13,14)(20,28)(24,29)(26,32)(30,31)(34,40)(35,43)(36,45)(42,44);(1,3,7,4)(2,8,11,9)(5,14,10,12)(6,13,16,15)(17,18)(19,23)(20,29)(21,25)(22,27)(24,28)(26,30)(31,32)(33,34)(35,41)(36,46)(37,45)(38,42)(39,40)(43,48)(44,47);(1,3,7,4)(2,9,11,8)(5,12,10,14)(6,13,16,15)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,32)(29,31)(33,35,39,43)(34,41,40,48)(36,38,45,47)(37,44,46,42);(1,4,7,3)(2,9,11,8)(5,14,10,12)(6,13,16,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,36)(34,37)(35,38)(39,45)(40,46)(41,44)(42,48)(43,47);(1,5,7,10)(2,6,11,16)(3,12,4,14)(8,15,9,13)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,29)(27,32)(33,37)(34,36)(35,44)(38,41)(39,46)(40,45)(42,43)(47,48);(1,6)(2,10)(3,13)(4,15)(5,11)(7,16)(8,12)(9,14)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,38)(34,42)(35,45)(36,43)(37,41)(39,47)(40,44)(46,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8380046);