#323019

order	512
nilpotency class	3
isoclinism	isoclinic class of #289819
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=hi,d^2=ei,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230215);