#8922698

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

4614 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gh,d^2=i,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=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, 8829894);