#240533

order	512
nilpotency class	3
isoclinism	isoclinic class of #233152
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

1012 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,g^b=gh,d^c=dhi,f^c=fhi

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, 147729);