#8843462

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3896 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8750658);