#120121

order	512
nilpotency class	3
isoclinism	isoclinic class of #100075
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

658 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=dh,e^2=i,g^2=hi,b^a=bd,c^a=ce,e^a=eh,f^a=fh,c^b=cf,e^b=eh,e^c=ei,g^c=ghi

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, 27317);