#7732828

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

4270 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)(9,10,14,11)(12,17,20,15)(13,16,21,18)(19,23,24,22)(25,26)(27,32)(28,33)(29,34)(30,31)(35,39)(36,37)(38,40);(1,3,5,8)(2,6,4,7)(26,31)(27,35)(29,33)(36,38);(1,4,5,2)(3,7,8,6)(10,11)(15,17)(16,18)(22,23)(25,27,30,35)(26,32,31,39)(28,38,34,36)(29,37,33,40);(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22);(9,12,14,20)(10,15,11,17)(13,19,21,24)(16,22,18,23)(25,28,30,34)(26,33,31,29)(27,36,35,38)(32,37,39,40);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29,30,33)(26,34,31,28)(27,37,35,40)(32,36,39,38)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=g,d^2=h,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, 7640024);