#358237

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2034 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=e,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=di,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265433);