#281717

order	512
nilpotency class	3
isoclinism	isoclinic class of #281659
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	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

824 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)(5,6,9,7,10,12,8,11)(13,14,19,25,18,16,17,21)(15,20,26,28,24,22,23,27);(2,4)(5,7,8,6,10,11,9,12)(13,15,19,26,18,24,17,23)(14,20,25,28,16,22,21,27);(6,7)(8,9)(11,12)(13,16,18,14)(15,22,24,20)(17,21,19,25)(23,27,26,28);(2,4)(13,17,18,19)(14,21,16,25)(15,23,24,26)(20,27,22,28)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 188913);