#412822

order	512
nilpotency class	3
isoclinism	isoclinic class of #393988
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1036 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,d^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^c=ei,f^c=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, 320018);