#8197150

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	#54862	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4446 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8104346);