#8462019

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	#42478	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4575 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=g,d^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8369215);