#120737

order	512
nilpotency class	3
isoclinism	isoclinic class of #10637
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#397
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

518 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=di,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,f^b=fi,e^c=eh,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, 27933);