#10171833

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4017 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)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(17,18)(19,28)(20,30)(21,27)(22,24)(23,25)(26,31)(29,32)(33,34)(35,44)(36,42)(37,38)(39,41)(40,43)(45,47)(46,48);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(17,19,23,22)(18,24,25,28)(20,29,31,27)(21,26,32,30)(33,35,40,39)(34,41,43,44)(36,45,37,46)(38,47,42,48);(2,4)(3,10)(9,12)(11,16)(18,25)(20,31)(24,28)(27,29)(33,36,40,37)(34,42,43,38)(35,45,39,46)(41,48,44,47);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,20)(18,26)(19,29)(21,28)(22,27)(23,31)(24,32)(25,30)(33,37,40,36)(34,42,43,38)(35,46,39,45)(41,48,44,47);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,21,23,32)(18,27,25,29)(19,30,22,26)(20,24,31,28)(33,38,40,42)(34,37,43,36)(35,47,39,48)(41,46,44,45);(1,6,7,14)(2,9,4,12)(3,11,10,16)(5,13,8,15)(17,22,23,19)(18,28,25,24)(20,27,31,29)(21,30,32,26)(33,39,40,35)(34,41,43,44)(36,46,37,45)(38,47,42,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=gh,d^2=ghi,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, 10079029);