#6370521

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

2274 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)(13,14,19,17,18,21,16,26)(15,22,23,28,24,20,27,25);(1,2,3,4)(6,9)(7,11)(13,15)(14,20)(16,27)(17,25)(18,24)(19,23)(21,22)(26,28);(5,7,8,11)(6,10,9,12)(13,16,18,19)(14,17,21,26)(15,23,24,27)(20,28,22,25);(6,9)(10,12)(14,21)(15,24)(17,26)(23,27);(13,17,19,21,18,26,16,14)(15,25,23,22,24,28,27,20)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,e^2=g,g^2=i,b^a=bf,c^a=ci,d^a=df,e^a=ei,c^b=cf,d^b=di,e^c=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, 6277717);