#6369999

order	512
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁶×C₄²

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	C₂²×#41	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2512 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)(7,10)(8,9)(11,12)(13,14,18,22,17,19,27,24)(15,20,25,28,23,16,21,26);(1,2,3,4)(6,9)(10,12)(14,19)(15,23)(21,25)(22,24);(13,15,18,25,17,23,27,21)(14,20,22,28,19,16,24,26);(5,7,8,11)(6,10,9,12)(13,16,17,20)(14,21,19,25)(15,24,23,22)(18,26,27,28);(14,19)(16,20)(22,24)(26,28)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=f,g^2=i,b^a=bf,e^a=ei,c^b=ci,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, 6277195);