#7883010

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	#59013	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4588 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,d^2=h,e^2=h,f^2=gi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7790206);