#393044

order	512
nilpotency class	3
isoclinism	isoclinic class of #32878
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2288 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fi,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^b=ei,h^b=hi

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, 300240);