#7904571

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	#39884	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4540 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7811767);