#10460584

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

4165 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10367780);