#282541

order	512
nilpotency class	3
isoclinism	isoclinic class of #281915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

937 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)(5,6,9,7,10,11,8,12)(13,14,16,20,17,21,22,19)(15,18,25,26,23,27,24,28);(2,4)(5,7,8,6,10,12,9,11)(13,15,22,24,17,23,16,25)(14,18,19,28,21,27,20,26);(6,7)(8,9)(11,12)(14,19)(15,23)(16,22)(18,26)(20,21)(27,28);(2,4)(6,11)(7,12)(13,16,17,22)(14,19,21,20)(15,24,23,25)(18,26,27,28)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,d^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=dh,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, 189737);