#288023

order	512
nilpotency class	3
isoclinism	isoclinic class of #282171
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#28448	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

807 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=ef,c^2=i,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 195219);