#7870447

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	#44069	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4335 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7777643);