#8688349

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	#42581	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4411 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8595545);