#10324689

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

3955 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)(3,11)(4,13)(5,10)(6,8)(7,14)(9,15)(12,16)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28)(33,34,38,43)(35,39,44,40)(36,41,46,47)(37,45,48,42);(1,3,6,14)(2,7,8,11)(4,10,15,12)(5,13,16,9)(19,27)(21,31)(23,32)(25,28)(33,35)(34,39)(36,45)(37,47)(38,44)(40,43)(41,48)(42,46);(2,8)(5,16)(7,11)(9,13)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,35,38,44)(34,40,43,39)(36,42,46,45)(37,41,48,47);(1,4,6,15)(2,9,8,13)(3,12,14,10)(5,7,16,11)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,36)(34,41)(35,42)(37,39)(38,46)(40,48)(43,47)(44,45);(1,5,6,16)(2,10,8,12)(3,13,14,9)(4,7,15,11)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,35)(34,39)(36,42)(37,41)(38,44)(40,43)(45,46)(47,48);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,16)(10,15)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)(33,37)(34,42)(35,41)(36,39)(38,48)(40,46)(43,45)(44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=g,e^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, 10231885);