#10424678

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	#2631	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3974 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=ghi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,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, 10331874);