#8928679

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	#2539	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8835875);