#8393315

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	#55044	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4137 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)(3,11)(4,13)(5,10)(6,7)(8,14)(9,15)(12,16)(33,34)(35,40)(36,37)(38,41)(39,42)(43,47)(44,45)(46,48);(2,7)(4,15)(8,11)(10,12)(17,18,22,23)(19,25,28,24)(20,29,31,21)(26,32,30,27)(33,34,39,42)(35,40,36,37)(38,41,46,48)(43,47,44,45);(2,7)(5,16)(8,11)(9,13)(18,23)(21,29)(24,25)(27,32)(33,35,39,36)(34,40,42,37)(38,44,46,43)(41,45,48,47);(1,3,6,14)(2,8,7,11)(4,10,15,12)(5,13,16,9)(17,19,22,28)(18,24,23,25)(20,30,31,26)(21,27,29,32)(33,36,39,35)(34,40,42,37)(38,44,46,43)(41,47,48,45);(1,4,6,15)(2,9,7,13)(3,12,14,10)(5,8,16,11)(17,20)(18,21)(19,26)(22,31)(23,29)(24,32)(25,27)(28,30)(33,37,39,40)(34,36,42,35)(38,45,46,47)(41,44,48,43);(1,5,6,16)(2,10,7,12)(3,13,14,9)(4,8,15,11)(17,21)(18,20)(19,27)(22,29)(23,31)(24,30)(25,26)(28,32)(33,38,39,46)(34,41,42,48)(35,43,36,44)(37,45,40,47)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300511);