#7659012

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4278 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)(10,16)(12,21)(17,18)(19,23)(25,26,30,29)(27,33,35,31)(28,36,37,32)(34,38,40,39);(1,2,4,5)(3,6,7,8)(9,10,15,16)(11,18,13,17)(12,22,21,14)(19,20,23,24)(25,27,30,35)(26,31,29,33)(28,34,37,40)(32,38,36,39);(9,11,15,13)(10,17,16,18)(12,19,21,23)(14,20,22,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,38)(36,39)(37,40);(9,12)(10,14)(11,19)(13,23)(15,21)(16,22)(17,20)(18,24)(25,28,30,37)(26,32,29,36)(27,34,35,40)(31,38,33,39);(1,3,4,7)(2,6,5,8)(9,13,15,11)(10,17,16,18)(12,23,21,19)(14,20,22,24)(25,29,30,26)(27,33,35,31)(28,36,37,32)(34,39,40,38);(9,14)(10,12)(11,20)(13,24)(15,22)(16,21)(17,19)(18,23)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=hi,d^2=h,e^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, 7566208);