#282024

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	#25599	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

849 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)(5,6,10,11,9,7,8,12)(13,14,17,21,18,22,16,20)(15,19,24,26,23,27,25,28);(2,4)(5,7,8,11,9,6,10,12)(13,15,17,24,18,23,16,25)(14,19,21,26,22,27,20,28);(5,6,9,7)(8,11,10,12)(14,20)(15,23)(16,17)(19,26)(21,22)(27,28);(2,4)(5,8,9,10)(6,11,7,12)(13,16,18,17)(14,20,22,21)(15,24,23,25)(19,26,27,28)

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,c^2=hi,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,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, 189220);