#7659016

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4162 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7566212);