#256494

order	512
nilpotency class	3
isoclinism	isoclinic class of #249259
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

785 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=e,d^2=hi,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,d^c=dhi,f^c=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 163690);