#7770290

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	#2403	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4630 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=ghi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=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, 7677486);