#8425590

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	#38938	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4439 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=g,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, 8332786);