#8406499

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	#55161	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3855 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)(4,7,12,14)(10,15,16,13)(17,18)(19,24)(20,23)(21,26)(22,25)(27,31)(28,30)(29,32)(33,34,39,38)(35,42,37,40)(36,41,44,46)(43,48,45,47);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19)(18,24)(20,26)(21,23)(22,27)(25,31)(28,32)(29,30)(36,44)(41,46)(43,45)(47,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,18,23,20)(19,26,21,24)(22,28,30,25)(27,31,29,32)(33,35,39,37)(34,40,38,42)(36,43,44,45)(41,47,46,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,20,23,18)(19,26,21,24)(22,28,30,25)(27,32,29,31)(33,36,39,44)(34,41,38,46)(35,43,37,45)(40,47,42,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,37,39,35)(34,42,38,40)(36,45,44,43)(41,48,46,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(33,38,39,34)(35,42,37,40)(36,46,44,41)(43,48,45,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8313695);