#7656813

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4424 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:

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

Polycyclic presentation:

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

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, 7564009);