#404407

order	512
nilpotency class	3
isoclinism	isoclinic class of #394289
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1199 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)(13,14,18,23,19,24,20,22)(15,25,26,17,27,28,21,16);(5,6)(7,10)(8,9)(11,12)(13,15,20,21,19,27,18,26)(14,17,23,28,24,16,22,25);(2,4)(6,9)(10,12)(13,16,19,17)(14,21,24,26)(15,23,27,22)(18,28,20,25);(5,7,8,11)(6,10,9,12)(13,17,19,16)(14,21,24,26)(15,23,27,22)(18,25,20,28)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 311603);