#10409772

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	#38013	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4139 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=gh,e^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, 10316968);