#7787748

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	#43526	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4346 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)(9,10,14,15)(11,16,17,12)(13,18,21,22)(19,23,24,20)(25,26,30,29)(27,33,34,31)(28,32,36,37)(35,39,40,38);(1,3,5,8)(2,6,4,7)(9,10,14,15)(11,16,17,12)(13,18,21,22)(19,23,24,20)(25,27)(26,31)(28,35)(29,33)(30,34)(32,38)(36,40)(37,39);(10,15)(11,17)(13,21)(20,23)(27,34)(31,33)(35,40)(38,39);(1,2,5,4)(3,6,8,7)(9,11)(10,16)(12,15)(13,19)(14,17)(18,23)(20,22)(21,24)(25,28,30,36)(26,32,29,37)(27,35,34,40)(31,38,33,39);(1,4,5,2)(3,7,8,6)(9,12,14,16)(10,17,15,11)(13,20,21,23)(18,24,22,19)(25,29,30,26)(27,31,34,33)(28,32,36,37)(35,39,40,38);(9,13)(10,18)(11,19)(12,20)(14,21)(15,22)(16,23)(17,24)(25,27)(26,33)(28,35)(29,31)(30,34)(32,39)(36,40)(37,38)

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=g,e^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=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, 7694944);