#6370382

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×#532	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2432 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)(7,11)(8,10)(9,12)(15,21)(18,24)(22,27)(25,28);(5,7)(6,9)(8,12)(10,11)(13,14,16,20,17,19,23,26)(15,18,22,25,21,24,27,28);(7,12)(9,11)(13,15)(14,18)(16,22)(17,21)(19,24)(20,25)(23,27)(26,28);(6,10)(9,11)(15,21)(18,24)(22,27)(25,28);(1,3,4,2)(14,19)(18,24)(20,26)(25,28)

Polycyclic presentation:

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

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, 6277578);