#205202

order	512
nilpotency class	3
isoclinism	isoclinic class of #19186
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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

1102 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 112398);