#7824387

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4293 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)(9,10,14,15)(11,12,17,20)(13,16,21,22)(18,19,23,24)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,40)(38,39);(1,2,5,6)(3,7,4,8)(25,27,30,32)(26,28,31,36)(29,37,38,34)(33,40,39,35);(1,3,5,4)(2,7,6,8)(10,15)(11,17)(16,22)(18,23)(26,31)(27,32)(29,38)(35,40);(1,3,5,4)(2,7,6,8)(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,40)(38,39);(1,3,5,4)(2,7,6,8)(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,28)(26,32)(27,31)(29,35)(30,36)(33,37)(34,39)(38,40);(1,4,5,3)(2,8,6,7)(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,29,30,38)(26,33,31,39)(27,34,32,37)(28,35,36,40)

Polycyclic presentation:

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

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, 7731583);