#10202123

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	#41061	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3992 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)(3,11,14,8)(4,15,13,9)(6,10,12,16)(17,18)(19,23)(20,24)(21,32)(22,26)(25,29)(27,28)(30,31)(33,34)(35,43)(36,40)(37,46)(38,42)(39,45)(41,48)(44,47);(1,3)(2,8)(4,12)(5,11)(6,13)(7,14)(9,16)(10,15)(17,19,22,28)(18,23,26,27)(20,29,31,21)(24,25,30,32)(33,34,38,42)(35,39,45,43)(36,44,47,40)(37,41,48,46);(17,18,22,26)(19,27,28,23)(20,24,31,30)(21,25,29,32)(33,35,38,45)(34,39,42,43)(36,41,47,46)(37,44,48,40);(1,4,7,13)(2,9,5,15)(3,12,14,6)(8,16,11,10)(17,18)(19,23)(20,30)(21,25)(22,26)(24,31)(27,28)(29,32)(33,35)(34,39)(36,46)(37,44)(38,45)(40,48)(41,47)(42,43);(1,5,7,2)(3,11,14,8)(4,9,13,15)(6,10,12,16)(17,20,22,31)(18,24,26,30)(19,21,28,29)(23,32,27,25)(33,36)(34,40)(35,41)(37,39)(38,47)(42,44)(43,48)(45,46);(1,6,7,12)(2,10,5,16)(3,13,14,4)(8,15,11,9)(17,21)(18,25)(19,20)(22,29)(23,30)(24,27)(26,32)(28,31)(33,37)(34,41)(35,44)(36,43)(38,48)(39,47)(40,45)(42,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=ghi,e^2=gh,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, 10109319);