#8772773

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	#42063	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4075 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)(3,7)(4,11)(5,6)(8,13)(9,16)(10,14)(12,15)(19,22)(24,27)(29,31)(30,32)(33,34,39,36)(35,42,41,38)(37,40,46,44)(43,48,47,45);(2,6)(7,14)(8,11)(15,16)(17,18,23,28)(19,27,22,24)(20,25,26,21)(29,30,31,32)(33,34,39,36)(35,38,41,42)(37,44,46,40)(43,48,47,45);(1,3,5,10)(2,7,6,14)(4,12,13,9)(8,16,11,15)(17,19,23,22)(18,24,28,27)(20,31,26,29)(21,32,25,30)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(1,4,5,13)(2,8,6,11)(3,9,10,12)(7,15,14,16)(17,20,23,26)(18,25,28,21)(19,29,22,31)(24,32,27,30)(33,36,39,34)(35,42,41,38)(37,44,46,40)(43,48,47,45);(1,3)(2,7)(4,9)(5,10)(6,14)(8,15)(11,16)(12,13)(17,21,23,25)(18,26,28,20)(19,30,22,32)(24,31,27,29)(33,37,39,46)(34,40,36,44)(35,43,41,47)(38,45,42,48);(1,2)(3,7)(4,8)(5,6)(9,15)(10,14)(11,13)(12,16)(17,22,23,19)(18,27,28,24)(20,31,26,29)(21,32,25,30)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gi,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8679969);