#10168448

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	#50053	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3993 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=i,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=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, 10075644);