#6370308

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2712 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,g^2=i,b^a=bf,d^a=df,e^a=ei,c^b=cf,e^b=ei,e^d=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277504);