#8471837

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4174 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)(3,9)(4,14)(5,13)(6,11)(7,8)(10,15)(12,16)(20,28)(24,29)(26,32)(30,31)(33,34)(35,40)(36,37)(38,42)(39,43)(41,47)(44,48)(45,46);(2,8)(5,15)(9,14)(11,12)(17,18)(19,23)(20,29)(21,25)(22,27)(24,28)(26,30)(31,32)(33,35,39,45)(34,40,43,46)(36,44,47,42)(37,48,41,38);(1,3,7,4)(2,9,8,14)(5,12,15,11)(6,13,16,10)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,32)(29,31)(33,36)(34,41)(35,42)(37,43)(38,46)(39,47)(40,48)(44,45);(1,4,7,3)(2,9,8,14)(5,11,15,12)(6,13,16,10)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,37,39,41)(34,36,43,47)(35,38,45,48)(40,42,46,44);(1,5)(2,10)(3,12)(4,11)(6,9)(7,15)(8,13)(14,16)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,29)(27,32)(33,38)(34,42)(35,41)(36,46)(37,45)(39,48)(40,47)(43,44);(1,6)(2,11)(3,13)(4,10)(5,9)(7,16)(8,12)(14,15)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,36)(34,37)(35,44)(38,46)(39,47)(40,48)(41,43)(42,45)

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

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, 8379033);