#10171825

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3925 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)(3,10)(4,7)(5,9)(6,8)(11,16)(12,14)(13,15)(17,18)(19,23)(20,24)(21,32)(22,26)(25,29)(27,28)(30,31)(34,40)(36,46)(41,43)(44,47);(1,3,7,6)(2,8,4,10)(5,13,14,11)(9,16,12,15)(17,19,22,28)(18,23,26,27)(20,29,31,21)(24,25,30,32)(33,34,39,40)(35,41,37,43)(36,42,46,38)(44,48,47,45);(2,4)(8,10)(9,12)(15,16)(17,18,22,26)(19,27,28,23)(20,24,31,30)(21,25,29,32)(33,35,39,37)(34,41,40,43)(36,44,46,47)(38,45,42,48);(1,4,7,2)(3,10,6,8)(5,9,14,12)(11,15,13,16)(17,18)(19,23)(20,30)(21,25)(22,26)(24,31)(27,28)(29,32)(33,36)(34,42)(35,44)(37,47)(38,40)(39,46)(41,48)(43,45);(1,5,7,14)(2,9,4,12)(3,11,6,13)(8,15,10,16)(17,20,22,31)(18,24,26,30)(19,21,28,29)(23,32,27,25)(33,37,39,35)(34,43,40,41)(36,44,46,47)(38,45,42,48);(1,6,7,3)(2,10,4,8)(5,11,14,13)(9,15,12,16)(17,21)(18,25)(19,20)(22,29)(23,30)(24,27)(26,32)(28,31)(33,38,39,42)(34,36,40,46)(35,45,37,48)(41,44,43,47)

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10079021);