#7941685

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	#35353	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4441 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,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, 7848881);