#10444975

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3936 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)(3,4)(5,9)(6,10)(7,8)(11,12)(13,15)(14,16)(17,18,23,20)(19,24,21,26)(22,25,30,28)(27,31,29,32)(33,34)(35,40)(36,38)(37,44)(39,43)(41,45)(42,46)(47,48);(1,2,6,10)(3,7,8,4)(5,9,14,16)(11,15,13,12)(17,19)(18,24)(20,26)(21,23)(22,27)(25,31)(28,32)(29,30)(33,34,38,36)(35,40,43,39)(37,41,45,44)(42,47,48,46);(17,18,23,20)(19,26,21,24)(22,25,30,28)(27,32,29,31)(33,35)(34,39)(36,40)(37,42)(38,43)(41,46)(44,47)(45,48);(1,3)(2,7)(4,10)(5,13)(6,8)(9,12)(11,14)(15,16)(17,20,23,18)(19,26,21,24)(22,25,30,28)(27,31,29,32)(33,36,38,34)(35,40,43,39)(37,44,45,41)(42,47,48,46);(1,4,6,7)(2,8,10,3)(5,12,14,15)(9,13,16,11)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,13)(10,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,37,38,45)(34,41,36,44)(35,42,43,48)(39,46,40,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=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, 10352171);