#8436912

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3869 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)(4,7,13,16)(10,15,12,14)(17,18,22,26)(19,21,25,23)(20,24,29,32)(27,28,31,30)(34,40)(35,36)(42,47)(43,44);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,19,22,25)(18,21,26,23)(20,27,29,31)(24,28,32,30)(33,34)(35,41)(36,38)(37,42)(39,40)(43,48)(44,45)(46,47);(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,35,39,36)(34,41,40,38)(37,43,46,44)(42,48,47,45);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,21)(20,27)(22,25)(23,26)(24,28)(29,31)(30,32)(33,36,39,35)(34,41,40,38)(37,44,46,43)(42,48,47,45);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,29)(18,24,26,32)(19,27,25,31)(21,28,23,30)(33,37,39,46)(34,42,40,47)(35,43,36,44)(38,45,41,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,21)(18,25)(19,26)(20,28)(22,23)(24,31)(27,32)(29,30)(33,38,39,41)(34,36,40,35)(37,45,46,48)(42,44,47,43)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=hi,d^2=g,e^2=h,f^2=g,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, 8344108);