#10451627

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	#2676	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4157 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:

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

Polycyclic presentation:

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

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, 10358823);