#10504993

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

4077 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=gh,e^2=i,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, 10412189);