#411836

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

2204 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

(2,4)(9,10,16,11)(12,20,22,24)(13,14,18,15)(17,23,19,21);(1,2,3,4)(5,6)(7,8)(9,11,14,18,16,10,15,13)(12,19,21,24,22,17,23,20);(6,8)(9,12)(10,17)(11,19)(13,20)(14,21)(15,23)(16,22)(18,24);(9,13,14,11,16,18,15,10)(12,20,21,19,22,24,23,17)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319032);