#7646238

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	#39853	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4473 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=gh,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7553434);