#8455666

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	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3939 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=hi,d^2=h,e^2=ghi,f^2=h,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, 8362862);