#315708

order	512
nilpotency class	3
isoclinism	isoclinic class of #298711
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

652 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=fh,d^2=ei,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 222904);