#392639

order	512
nilpotency class	3
isoclinism	isoclinic class of #32884
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3924 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)(7,8)(9,10)(11,12)(13,14,17,21,18,22,16,20)(15,23,26,19,24,28,25,27);(1,2,3,4)(13,15,18,24)(14,19,22,27)(16,26,17,25)(20,28,21,23);(6,8)(13,14,18,22)(15,19,24,27)(16,21,17,20)(23,25,28,26);(10,12)(15,24)(19,27)(23,28)(25,26)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,d^b=di,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi

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, 299835);