#8445135

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	#42041	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4087 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)(3,10)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(20,28)(24,31)(27,32)(29,30)(34,40)(35,44)(36,46)(43,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,18,22,26)(19,21,25,23)(20,24,28,31)(27,29,32,30)(33,34)(35,41)(36,47)(37,43)(38,44)(39,40)(42,46)(45,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,35,39,44)(34,41,40,38)(36,45,46,37)(42,43,47,48);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(17,19)(18,21)(20,27)(22,25)(23,26)(24,29)(28,32)(30,31)(33,36)(34,42)(35,45)(37,44)(38,48)(39,46)(40,47)(41,43);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,20,22,28)(18,24,26,31)(19,27,25,32)(21,29,23,30)(33,37,39,45)(34,43,40,48)(35,36,44,46)(38,42,41,47);(1,2)(3,7)(4,8)(5,14)(6,9)(10,13)(11,15)(12,16)(17,21)(18,25)(19,26)(20,29)(22,23)(24,32)(27,31)(28,30)(33,38,39,41)(34,44,40,35)(36,43,46,48)(37,42,45,47)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=gh,e^2=hi,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, 8352331);