#415908

order	512
nilpotency class	3
isoclinism	isoclinic class of #394192
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1636 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=eh,d^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=di,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323104);