#8657953

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	#38564	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4566 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi,f^e=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, 8565149);