#8976415

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

4010 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:

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

Polycyclic presentation:

2^9|a^2=gh,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, 8883611);