#7849000

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	#52165	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4531 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=i,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7756196);