#7732817

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	#37446	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4694 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)(9,10,14,15)(11,16,18,13)(12,20,22,17)(19,21,24,23)(25,26)(27,29)(28,35)(30,31)(32,36)(33,38)(34,40)(37,39);(1,3,5,8)(2,6,4,7)(25,27,30,33)(26,29,31,38)(28,34,36,39)(32,37,35,40);(1,4,5,2)(3,7,8,6)(10,15)(11,18)(17,20)(19,24)(26,31)(27,33)(28,36)(37,40);(9,11)(10,16)(12,19)(13,15)(14,18)(17,23)(20,21)(22,24)(25,26)(27,29)(28,32)(30,31)(33,38)(34,37)(35,36)(39,40);(9,12,14,22)(10,17,15,20)(11,19,18,24)(13,21,16,23)(25,28)(26,32)(27,34)(29,37)(30,36)(31,35)(33,39)(38,40);(9,13,14,16)(10,11,15,18)(12,21,22,23)(17,19,20,24)(25,29)(26,27)(28,37)(30,38)(31,33)(32,34)(35,39)(36,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=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, 7640013);