#8425402

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4014 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)(3,9)(4,7)(5,8)(6,10)(11,16)(12,14)(13,15)(18,24)(20,25)(26,32)(28,30)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(2,4)(8,9)(10,12)(15,16)(17,18,23,24)(19,25,21,20)(22,26,31,32)(27,30,29,28)(35,42)(39,46)(43,47)(44,48);(1,3,7,5)(2,8,4,9)(6,11,14,13)(10,15,12,16)(17,19,23,21)(18,25,24,20)(22,27,31,29)(26,30,32,28)(33,35,38,42)(34,39,41,46)(36,43,40,47)(37,44,45,48);(1,4,7,2)(3,9,5,8)(6,10,14,12)(11,15,13,16)(17,20)(18,19)(21,24)(22,30)(23,25)(26,29)(27,32)(28,31)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,5,7,3)(2,9,4,8)(6,13,14,11)(10,16,12,15)(17,21,23,19)(18,20,24,25)(22,29,31,27)(26,28,32,30);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,32)(25,30)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332598);