#10529836

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	#48505	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4084 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=g,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,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, 10437032);