#8247597

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	#59650	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4403 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8154793);