#7732830

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	#44236	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4434 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,d^2=gh,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, 7640026);