#6467453

order	512
nilpotency class	3
isoclinism	isoclinic class of #24264
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=hi,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6374649);