#10440029

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

4127 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)(3,9)(4,8)(5,7)(6,13)(10,14)(11,15)(12,16)(17,18)(19,25)(20,24)(21,23)(22,26)(27,32)(28,31)(29,30)(33,34,39,42)(35,40,37,45)(36,38,41,48)(43,44,46,47);(2,5)(8,9)(10,13)(15,16)(17,19,23,20)(18,24,21,25)(22,27,30,28)(26,31,29,32)(35,37)(40,45)(43,46)(44,47);(1,3,7,4)(2,8,5,9)(6,11,14,12)(10,15,13,16)(17,19,23,20)(18,25,21,24)(22,27,30,28)(26,32,29,31)(33,35,39,37)(34,40,42,45)(36,43,41,46)(38,44,48,47);(1,4,7,3)(2,9,5,8)(6,11,14,12)(10,15,13,16)(17,20,23,19)(18,25,21,24)(22,27,30,28)(26,31,29,32)(33,36)(34,38)(35,43)(37,46)(39,41)(40,44)(42,48)(45,47);(1,5,7,2)(3,9,4,8)(6,13,14,10)(11,16,12,15)(17,21,23,18)(19,24,20,25)(22,29,30,26)(27,31,28,32)(33,37,39,35)(34,40,42,45)(36,43,41,46)(38,47,48,44);(1,6,7,14)(2,10,5,13)(3,11,4,12)(8,15,9,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,38)(34,41)(35,44)(36,42)(37,47)(39,48)(40,46)(43,45)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=gi,d^2=g,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, 10347225);