#434218

order	512
nilpotency class	3
isoclinism	isoclinic class of #417581
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1442 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:

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

Polycyclic presentation:

2^9|a^2=eg,b^2=i,c^2=fi,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 341414);