#9945591

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

4100 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)(3,12)(4,9)(5,10)(6,8)(7,11)(13,14)(15,16)(17,18,22,20)(19,25,27,23)(21,28,29,24)(26,30,32,31)(33,34,39,43)(35,44,46,37)(36,41,48,40)(38,47,45,42);(1,3,7,6)(2,8,11,12)(4,15,10,13)(5,16,9,14)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30)(33,35)(34,37)(36,38)(39,46)(40,47)(41,42)(43,44)(45,48);(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,36,39,48)(34,40,43,41)(35,38,46,45)(37,47,44,42);(1,4,7,10)(2,9,11,5)(3,13,6,15)(8,16,12,14)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,36)(34,41)(35,45)(37,47)(38,46)(39,48)(40,43)(42,44);(1,5,7,9)(2,10,11,4)(3,14,6,16)(8,15,12,13)(17,20,22,18)(19,25,27,23)(21,24,29,28)(26,30,32,31)(33,37)(34,35)(36,47)(38,40)(39,44)(41,45)(42,48)(43,46);(1,6,7,3)(2,8,11,12)(4,15,10,13)(5,14,9,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,45)(34,42,43,47)(35,36,46,48)(37,41,44,40)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9852787);