#10409780

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	#49389	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4134 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=gh,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, 10316976);