#10460582

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

4292 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10367778);