#7656999

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	#37554	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4460 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)(9,10,14,18)(11,16,21,15)(12,13,17,23)(19,22,24,20)(25,26)(27,36)(28,34)(29,30)(31,33)(32,35)(37,40)(38,39);(1,2,5,6)(3,7,4,8)(25,27,32,31)(26,33,35,36)(28,39,30,37)(29,38,34,40);(9,11)(10,15)(12,19)(13,20)(14,21)(16,18)(17,24)(22,23)(25,28,32,30)(26,34,35,29)(27,37,31,39)(33,38,36,40);(9,12,14,17)(10,13,18,23)(11,19,21,24)(15,20,16,22)(25,29)(26,28)(27,38)(30,35)(31,40)(32,34)(33,39)(36,37);(1,3,5,4)(2,7,6,8)(9,11)(10,16)(12,19)(13,22)(14,21)(15,18)(17,24)(20,23)(25,30,32,28)(26,29,35,34)(27,37,31,39)(33,38,36,40);(1,4,5,3)(2,8,6,7)(9,13)(10,17)(11,20)(12,18)(14,23)(15,24)(16,19)(21,22)(25,31,32,27)(26,33,35,36)(28,39,30,37)(29,40,34,38)

Polycyclic presentation:

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

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, 7564195);