#8322394

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3784 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:

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

Polycyclic presentation:

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

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, 8229590);