#7763288

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4332 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=hi,e^2=hi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi

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, 7670484);