#8464066

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	#39910	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4104 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)(3,12)(4,6)(5,15)(7,11)(8,16)(9,14)(10,13)(17,18)(19,24)(20,25)(21,26)(22,27)(23,28)(29,30)(31,32)(34,40)(35,36)(42,47)(43,44);(1,3)(2,8)(4,13)(5,14)(6,15)(7,16)(9,10)(11,12)(19,28)(23,24)(29,32)(30,31)(33,34)(35,41)(36,38)(37,42)(39,40)(43,48)(44,45)(46,47);(17,19)(18,23)(20,29)(21,30)(22,28)(24,27)(25,31)(26,32)(33,35,39,36)(34,41,40,38)(37,43,46,44)(42,48,47,45);(1,4,7,14)(2,9,11,6)(3,13,16,5)(8,10,12,15)(17,19)(18,24)(20,29)(21,31)(22,28)(23,27)(25,30)(26,32)(33,36,39,35)(34,41,40,38)(37,44,46,43)(42,48,47,45);(1,5,7,13)(2,10,11,15)(3,14,16,4)(6,8,9,12)(17,20,22,26)(18,25,27,21)(19,29,28,32)(23,31,24,30)(33,37,39,46)(34,42,40,47)(35,43,36,44)(38,45,41,48);(1,6,7,9)(2,4,11,14)(3,15,16,10)(5,8,13,12)(17,21,22,25)(18,26,27,20)(19,30,28,31)(23,32,24,29)(33,38,39,41)(34,36,40,35)(37,45,46,48)(42,44,47,43)

Polycyclic presentation:

2^9|c^2=hi,d^2=g,e^2=gi,f^2=gi,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, 8371262);