#8458704

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	#45601	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4145 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=ghi,f^2=i,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, 8365900);