#206262

order	512
nilpotency class	3
isoclinism	isoclinic class of #19713
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3207	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1204 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:

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

Polycyclic presentation:

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

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, 113458);