#357510

order	512
nilpotency class	3
isoclinism	isoclinic class of #1999
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂⁴	C₂²×D₄
prime radical	C₂³×#135	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4758 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)(5,6)(7,8)(9,10,11,12)(13,14)(15,23)(16,20)(17,21)(18,22)(19,25)(24,28)(26,27);(2,4)(13,15)(14,19)(16,24)(17,25)(18,26)(20,23)(21,27)(22,28);(6,8)(13,14)(15,23)(16,20)(17,21)(18,22)(19,25)(24,28)(26,27);(13,16,18,17)(14,20,22,21)(15,24,26,25)(19,23,28,27)

Polycyclic presentation:

2^9|a^2=h,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,e^c=ei,g^c=gi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264706);