#354181

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1312 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)(7,10)(8,9)(11,12)(13,14,19,23,18,22,17,21)(15,24,25,27,26,20,28,16);(1,2,3,4)(6,9)(10,12)(13,15)(14,20)(16,23)(17,28)(18,26)(19,25)(21,27)(22,24);(13,16,17,24,18,27,19,20)(14,15,23,28,22,26,21,25);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,21,22,23)(15,25,26,28)(16,24,27,20)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fgi,d^2=e,g^2=i,b^a=be,c^a=cf,f^a=fi,d^b=di,f^c=fi,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261377);