#8202462

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	#40420	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3999 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8109658);