#10408576

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

4302 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=g,d^2=h,e^2=g,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, 10315772);