#8976377

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	#2812	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4028 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,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, 8883573);