#10312385

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	#44181	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3917 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,e^d=egi,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, 10219581);