#10423027

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

4109 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)(8,10)(12,16)(18,24)(19,20)(22,26)(29,31)(33,34)(35,36)(37,40)(38,41)(39,43)(42,44)(45,46)(47,48);(1,2,6,3)(4,9,11,7)(5,8,13,10)(12,15,16,14)(17,18)(19,28)(20,25)(21,26)(22,27)(23,24)(29,30)(31,32)(35,39)(36,43)(42,47)(44,48);(17,19,23,20)(18,25,24,28)(21,29,27,31)(22,30,26,32)(33,35,38,39)(34,36,41,43)(37,42,46,47)(40,44,45,48);(1,3,6,2)(4,7,11,9)(5,8,13,10)(12,15,16,14)(17,20,23,19)(18,25,24,28)(21,29,27,31)(22,32,26,30)(33,34)(35,36)(37,45)(38,41)(39,43)(40,46)(42,48)(44,47);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,21,23,27)(18,26,24,22)(19,29,20,31)(25,32,28,30)(33,36)(34,39)(35,41)(37,44)(38,43)(40,47)(42,45)(46,48);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,22,23,26)(18,27,24,21)(19,30,20,32)(25,31,28,29)(33,37)(34,40)(35,42)(36,44)(38,46)(39,47)(41,45)(43,48)

Polycyclic presentation:

2^9|b^2=gh,c^2=i,d^2=g,e^2=g,f^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, 10330223);