#7906427

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4182 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)(11,20)(12,14)(13,23)(15,17)(16,21)(18,24)(19,22)(27,33)(29,37)(31,35)(32,36);(1,2)(3,6)(4,5)(7,8)(9,11,15,21)(10,16,17,20)(12,22,24,13)(14,23,18,19)(25,26,30,28)(27,31,33,35)(29,32,37,36)(34,38,39,40);(10,17)(11,21)(12,24)(19,23)(25,27)(26,31)(28,35)(29,34)(30,33)(32,38)(36,40)(37,39);(1,3,4,7)(2,6,5,8)(9,12)(10,18)(11,22)(13,21)(14,17)(15,24)(16,19)(20,23)(25,28,30,26)(27,31,33,35)(29,36,37,32)(34,38,39,40);(9,13,15,22)(10,19,17,23)(11,12,21,24)(14,20,18,16)(25,29)(26,32)(27,34)(28,36)(30,37)(31,38)(33,39)(35,40);(9,14,15,18)(10,12,17,24)(11,23,21,19)(13,16,22,20)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7813623);