#8425609

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4260 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332805);