#10195398

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3842 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,d^2=h,e^2=g,f^2=ghi,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, 10102594);