#8908122

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	#2678	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3984 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=h,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8815318);