#403402

order	512
nilpotency class	3
isoclinism	isoclinic class of #394048
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
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

2355 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=g,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,h^b=hi,f^c=fi,e^d=ei,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, 310598);