#10216995

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3897 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)(17,18,23,28)(19,27,22,24)(20,30,31,25)(21,26,32,29)(33,34)(35,44)(36,46)(37,41)(38,45)(39,40)(42,47)(43,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,15)(12,16)(13,14)(17,19,23,22)(18,24,28,27)(20,29,31,26)(21,30,32,25)(34,40)(35,37)(42,46)(43,45);(17,20,23,31)(18,25,28,30)(19,29,22,26)(21,27,32,24)(33,35,39,37)(34,41,40,44)(36,43,47,45)(38,42,48,46);(1,2)(3,9)(4,8)(5,6)(7,11)(10,16)(12,15)(13,14)(17,20)(18,25)(19,29)(21,24)(22,26)(23,31)(27,32)(28,30)(33,36,39,47)(34,42,40,46)(35,45,37,43)(38,44,48,41);(1,3,5,11)(2,7,6,9)(4,12,13,10)(8,16,14,15)(17,21)(18,26)(19,25)(20,27)(22,30)(23,32)(24,31)(28,29)(33,37,39,35)(34,41,40,44)(36,43,47,45)(38,46,48,42);(1,4,5,13)(2,8,6,14)(3,10,11,12)(7,15,9,16)(17,22,23,19)(18,27,28,24)(20,26,31,29)(21,25,32,30)(33,38)(34,43)(35,42)(36,44)(37,46)(39,48)(40,45)(41,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=i,e^2=ghi,f^2=gi,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, 10124191);