#203759

order	512
nilpotency class	3
isoclinism	isoclinic class of #19254
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

798 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,e^b=eh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 110955);