#10324069

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

4370 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=h,e^2=h,f^2=gh,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, 10231265);