#8258572

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	#58340	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3978 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)(3,8)(4,10)(5,12)(6,11)(7,9)(13,16)(14,15)(17,18,24,28)(19,29,23,25)(20,31,30,26)(21,32,27,22)(33,34,38,37)(35,39,41,45)(36,40,43,44)(42,46,47,48);(1,3)(2,8)(4,6)(5,9)(7,12)(10,11)(13,14)(15,16)(17,19,24,23)(18,25,28,29)(20,27,30,21)(22,31,32,26)(35,41)(36,43)(39,45)(40,44);(2,9)(3,12)(6,13)(10,15)(17,20)(18,26)(19,21)(22,29)(23,27)(24,30)(25,32)(28,31)(33,35,38,41)(34,39,37,45)(36,42,43,47)(40,46,44,48);(1,4)(2,10)(3,13)(5,11)(6,12)(7,14)(8,16)(9,15)(17,21,24,27)(18,22,28,32)(19,30,23,20)(25,26,29,31);(1,5)(2,3)(4,11)(6,15)(7,8)(9,12)(10,13)(14,16)(17,22)(18,27)(19,31)(20,29)(21,28)(23,26)(24,32)(25,30)(33,36,38,43)(34,40,37,44)(35,42,41,47)(39,46,45,48);(1,6,7,13)(2,11,9,16)(3,4,12,14)(5,15,8,10)(17,23,24,19)(18,25,28,29)(20,27,30,21)(22,26,32,31)(33,37,38,34)(35,39,41,45)(36,44,43,40)(42,46,47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=g,e^2=hi,f^2=i,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, 8165768);