#8458809

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

4201 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:

(2,6)(3,9)(8,11)(10,15)(17,18,22,21)(19,25,27,23)(20,28,29,24)(26,30,32,31)(36,44)(40,47)(43,48)(45,46);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(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)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(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);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(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);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(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=h,b^2=h,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, 8366005);