#10247075

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	#48707	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4094 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)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18,23,27)(19,28,21,24)(20,29,31,25)(22,26,32,30)(33,34,39,41)(35,44,45,40)(36,46,43,38)(37,42,47,48);(4,11)(8,14)(10,15)(12,16)(17,19,23,21)(18,24,27,28)(20,30,31,26)(22,29,32,25)(33,35)(34,40)(36,37)(38,42)(39,45)(41,44)(43,47)(46,48);(2,7)(3,9)(8,14)(10,15)(34,41)(35,45)(36,43)(42,48);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,20,23,31)(18,25,27,29)(19,26,21,30)(22,24,32,28)(33,36,39,43)(34,38,41,46)(35,37,45,47)(40,42,44,48);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,21,23,19)(18,24,27,28)(20,26,31,30)(22,29,32,25)(33,37)(34,42)(35,43)(36,45)(38,44)(39,47)(40,46)(41,48);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,22)(18,26)(19,29)(20,24)(21,25)(23,32)(27,30)(28,31)(33,38)(34,43)(35,42)(36,41)(37,44)(39,46)(40,47)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154271);