#8094742

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	#45882	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4187 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=gh,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

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, 8001938);