#7728238

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4156 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gi,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei

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, 7635434);