#6370502

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
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×#121	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2126 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277698);