#10352533

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
prime radical	#37776	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4505 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=g,d^2=hi,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, 10259729);