#411968

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1190 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319164);