#7720001

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

4166 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)(10,15)(13,22)(16,23)(17,20)(25,26,31,29)(27,32,28,34)(30,37,38,33)(35,40,36,39);(1,2)(3,7)(4,8)(5,6)(9,10)(11,16)(12,20)(13,18)(14,15)(17,21)(19,23)(22,24)(25,27,31,28)(26,32,29,34)(30,35,38,36)(33,39,37,40);(10,15)(12,21)(16,23)(18,24)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(1,3,5,4)(2,7,6,8)(9,11,14,19)(10,16,15,23)(12,18,21,24)(13,17,22,20)(25,29,31,26)(27,34,28,32)(30,37,38,33)(35,40,36,39);(1,4,5,3)(2,8,6,7)(9,12,14,21)(10,17,15,20)(11,18,19,24)(13,16,22,23)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40);(9,13,14,22)(10,18,15,24)(11,17,19,20)(12,16,21,23)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,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, 7627197);