#10352522

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3821 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)(3,11)(4,9)(5,12)(6,8)(7,14)(10,16)(13,15)(17,18,22,20)(19,23,25,27)(21,24,30,28)(26,31,29,32)(33,34,40,43)(35,38,42,48)(36,45,47,41)(37,39,46,44);(1,3,6,14)(2,7,8,11)(4,12,15,10)(5,9,16,13)(17,19)(18,23)(20,27)(21,29)(22,25)(24,32)(26,30)(28,31)(33,35,40,42)(34,38,43,48)(36,44,47,39)(37,41,46,45);(2,8)(4,15)(7,11)(10,12)(19,25)(23,27)(26,29)(31,32)(33,36,40,47)(34,41,43,45)(35,44,42,39)(37,38,46,48);(1,3)(2,7)(4,12)(5,13)(6,14)(8,11)(9,16)(10,15)(17,20,22,18)(19,23,25,27)(21,28,30,24)(26,31,29,32)(33,37)(34,39)(35,41)(36,38)(40,46)(42,45)(43,44)(47,48);(1,4,6,15)(2,9,8,13)(3,12,14,10)(5,7,16,11)(17,19)(18,23)(20,27)(21,26)(22,25)(24,31)(28,32)(29,30)(33,38)(34,42)(35,43)(36,46)(37,47)(39,41)(40,48)(44,45);(1,5,6,16)(2,10,8,12)(3,13,14,9)(4,7,15,11)(17,21)(18,24)(19,26)(20,28)(22,30)(23,31)(25,29)(27,32)(33,39)(34,37)(35,36)(38,41)(40,44)(42,47)(43,46)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=i,e^2=g,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, 10259718);