#8295569

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	#56473	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3907 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=i,e^2=gh,f^2=g,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, 8202765);