#8324625

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	#58307	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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)(3,6,9,15)(4,5,8,14)(11,12,13,16)(17,18)(19,27)(20,22)(21,25)(23,24)(26,30)(28,29)(31,32)(33,34,40,37)(35,41,38,47)(36,45,44,42)(39,43,46,48);(1,3)(2,6)(4,11)(5,12)(7,9)(8,13)(10,15)(14,16)(18,24)(20,30)(25,27)(28,31)(33,35,40,38)(34,41,37,47)(36,39,44,46)(42,48,45,43);(3,9)(6,15)(11,13)(12,16)(17,19,23,21)(18,25,24,27)(20,28,30,31)(22,32,26,29)(33,36)(34,42)(35,39)(37,45)(38,46)(40,44)(41,48)(43,47);(1,4)(2,5)(3,11)(6,12)(7,8)(9,13)(10,14)(15,16)(17,20)(18,26)(19,28)(21,31)(22,24)(23,30)(25,29)(27,32)(33,37,40,34)(35,41,38,47)(36,45,44,42)(39,48,46,43);(1,5)(2,8)(3,12)(4,10)(6,13)(7,14)(9,16)(11,15)(17,21,23,19)(18,27,24,25)(20,28,30,31)(22,32,26,29)(33,38,40,35)(34,41,37,47)(36,46,44,39)(42,48,45,43);(1,6,7,15)(2,9,10,3)(4,12,8,16)(5,13,14,11)(17,22,23,26)(18,20,24,30)(19,29,21,32)(25,31,27,28)(33,39,40,46)(34,43,37,48)(35,44,38,36)(41,42,47,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=gh,e^2=h,f^2=hi,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, 8231821);