#8461301

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	#42448	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4097 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^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, 8368497);