#8458570

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	#45486	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4372 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8365766);