#6370216

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	C₂⁵	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2430 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,g^2=i,h^2=i,b^a=bf,d^a=df,c^b=cf,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, 6277412);