#296970

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

773 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:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=h,c^2=f,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,d^c=di

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 204166);