#299193

order	512
nilpotency class	3
isoclinism	isoclinic class of #299067
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25720	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1167 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=fhi,d^2=ehi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,d^c=dh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206389);