#8434336

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

4081 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8341532);