#7644112

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	#42674	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4228 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)(13,21)(17,22)(18,24)(20,23)(25,26,30,31)(27,34,29,32)(28,33,37,38)(35,40,36,39);(1,3,5,8)(2,6,4,7)(9,10)(11,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,24)(26,31)(32,34)(33,38)(39,40);(10,15)(12,16)(17,22)(20,23)(25,27,30,29)(26,32,31,34)(28,36,37,35)(33,40,38,39);(9,11)(10,16)(12,15)(13,18)(14,19)(17,23)(20,22)(21,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,40)(37,38);(9,12,14,16)(10,11,15,19)(13,20,21,23)(17,18,22,24)(25,28,30,37)(26,33,31,38)(27,35,29,36)(32,39,34,40);(1,4,5,2)(3,7,8,6)(9,13)(10,17)(11,18)(12,20)(14,21)(15,22)(16,23)(19,24)(25,29,30,27)(26,34,31,32)(28,36,37,35)(33,40,38,39)

Polycyclic presentation:

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

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, 7551308);