#10349046

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	#58583	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4086 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gi,d^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10256242);