#10322578

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	#36973	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4177 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=g,e^2=g,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, 10229774);