#7905973

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	#35361	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4617 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)(9,10,15,17)(11,14,20,16)(12,23,22,18)(13,24,21,19)(25,26)(27,31)(28,38)(29,33)(30,34)(32,36)(35,37)(39,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,21)(13,22)(14,17)(15,20)(18,19)(23,24)(25,26)(27,35)(28,32)(29,33)(30,34)(31,37)(36,38)(39,40);(10,17)(11,20)(12,22)(19,24)(25,27,30,37)(26,31,34,35)(28,29,36,40)(32,39,38,33);(1,3,4,7)(2,6,5,8)(9,12,15,22)(10,18,17,23)(11,21,20,13)(14,24,16,19)(25,28,30,36)(26,32,34,38)(27,29,37,40)(31,39,35,33);(9,13,15,21)(10,19,17,24)(11,22,20,12)(14,23,16,18)(25,29)(26,33)(27,36)(28,37)(30,40)(31,32)(34,39)(35,38);(9,14)(10,20)(11,17)(12,24)(13,18)(15,16)(19,22)(21,23)(25,26)(27,31)(28,32)(29,39)(30,34)(33,40)(35,37)(36,38)

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=ghi,e^2=h,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, 7813169);