#7720353

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4176 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,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, 7627549);