#8807458

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4553 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,d^2=h,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

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, 8714654);