#6370512

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

2310 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)(13,14,18,20,17,19,15,22)(16,21,24,27,25,26,23,28);(1,2,3,4)(6,9)(7,11)(14,19)(20,22)(21,26)(27,28);(5,7,8,11)(6,10,9,12)(13,15,17,18)(14,20,19,22)(16,24,25,23)(21,28,26,27);(6,9)(10,12)(14,19)(20,22)(21,26)(27,28);(13,16,18,24,17,25,15,23)(14,21,20,27,19,26,22,28)

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,c^b=cfi,e^c=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, 6277708);