#7917728

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4125 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)(11,21)(12,18)(13,24)(14,20)(15,17)(16,23)(19,22)(25,26,30,32)(27,33,36,31)(28,37,38,29)(34,35,40,39);(1,2)(3,7)(4,8)(5,6)(9,11)(10,16)(12,19)(13,18)(14,22)(15,23)(17,21)(20,24)(25,27)(26,31)(28,34)(29,35)(30,36)(32,33)(37,39)(38,40);(1,3,5,4)(2,7,6,8)(10,17)(12,20)(16,21)(19,24)(25,27,30,36)(26,31,32,33)(28,34,38,40)(29,35,37,39);(1,3,5,4)(2,7,6,8)(9,12,15,20)(10,18,17,14)(11,19,23,24)(13,21,22,16)(25,27)(26,31)(28,34)(29,35)(30,36)(32,33)(37,39)(38,40);(1,4,5,3)(2,8,6,7)(9,13,15,22)(10,19,17,24)(11,18,23,14)(12,21,20,16)(25,28,30,38)(26,29,32,37)(27,34,36,40)(31,35,33,39);(9,14,15,18)(10,20,17,12)(11,22,23,13)(16,24,21,19)(25,29)(26,28)(27,35)(30,37)(31,34)(32,38)(33,40)(36,39)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7824924);