#236041

order	512
nilpotency class	3
isoclinism	isoclinic class of #233936
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3036	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

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, 143237);