#8475175

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	#53470	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4245 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8382371);