#8425405

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

4002 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,d^2=gh,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, 8332601);