#7734376

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	#47851	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4487 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)(3,6,4,7)(9,10,15,18)(11,13,16,12)(14,17,23,24)(19,21,22,20)(25,26)(27,32)(28,33)(29,34)(30,35)(31,38)(36,40)(37,39);(1,2)(3,6)(4,7)(5,8)(9,11,15,16)(10,12,18,13)(14,22,23,19)(17,21,24,20)(25,27)(26,31)(28,36)(29,33)(30,38)(32,35)(34,39)(37,40);(1,3,5,4)(2,6,8,7)(9,12,15,13)(10,16,18,11)(14,20,23,21)(17,22,24,19)(25,27,30,38)(26,32,35,31)(28,34,39,36)(29,37,40,33);(1,3,5,4)(2,6,8,7)(9,13,15,12)(10,16,18,11)(14,21,23,20)(17,22,24,19)(25,28,30,39)(26,33,35,37)(27,36,38,34)(29,32,40,31);(9,10)(11,12)(13,16)(14,17)(15,18)(19,20)(21,22)(23,24)(25,27)(26,32)(28,36)(29,37)(30,38)(31,35)(33,40)(34,39);(1,4,5,3)(2,7,8,6)(9,14,15,23)(10,17,18,24)(11,19,16,22)(12,20,13,21)(25,29,30,40)(26,34,35,36)(27,37,38,33)(28,32,39,31)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=g,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7641572);