#10368641

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

4273 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)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,18,22,24)(19,27,29,23)(20,25,30,28)(21,31,32,26)(33,34)(35,40)(36,38)(37,44)(39,43)(41,45)(42,46)(47,48);(2,3)(5,13)(7,9)(12,16)(17,19,22,29)(18,23,24,27)(20,26,30,31)(21,25,32,28)(33,35)(34,39)(36,40)(37,42)(38,43)(41,46)(44,47)(45,48);(4,11)(7,9)(12,16)(14,15)(18,24)(20,30)(23,27)(26,31)(33,34,38,36)(35,40,43,39)(37,41,45,44)(42,47,48,46);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,19)(18,23)(20,26)(21,28)(22,29)(24,27)(25,32)(30,31)(33,36,38,34)(35,40,43,39)(37,44,45,41)(42,47,48,46);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20)(18,25)(19,26)(21,27)(22,30)(23,32)(24,28)(29,31)(33,35)(34,40)(36,39)(37,42)(38,43)(41,47)(44,46)(45,48);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21,22,32)(18,26,24,31)(19,28,29,25)(20,27,30,23)(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=hi,c^2=i,d^2=ghi,f^2=g,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, 10275837);