#10368342

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

4361 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,d^2=hi,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

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, 10275538);