#10324017

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4114 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=gh,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, 10231213);